Fast Finality: What Is Fast Finality in Cryptocurrency?Fast finality is the ability of a blockchain to make a transaction effectively irreversible within a short and predictable period.Finality is reached when the pFast Finality: What Is Fast Finality in Cryptocurrency?Fast finality is the ability of a blockchain to make a transaction effectively irreversible within a short and predictable period.Finality is reached when the p

Fast Finality

2026/08/10 11:29
#Intermediate

What Is Fast Finality in Cryptocurrency?

Fast finality is the ability of a blockchain to make a transaction effectively irreversible within a short and predictable period.

Finality is reached when the protocol provides sufficient assurance that the block containing the transaction will remain part of the accepted blockchain history.

A network with fast finality may settle transactions within one block, several voting rounds, a few seconds, or another relatively short interval.

The term does not have one universal technical threshold.

Different blockchain protocols use different definitions of when a transaction becomes final.

Some networks provide deterministic or Byzantine fault-tolerant finality after a supermajority of validators approves a block.

Some provide economic finality backed by validator stake and slashing penalties.

Others provide probabilistic finality, where the chance of reversal becomes extremely small after enough consensus activity.

A wallet or application may also describe a transaction as confirmed before it has reached the network’s strongest finality level.

Fast finality should therefore be evaluated by examining the exact consensus guarantee rather than relying only on a claimed settlement time.

What Does Blockchain Finality Mean?

Blockchain finality means that a transaction or block has become a permanent part of the accepted ledger under the protocol’s stated security assumptions.

Before finality, a competing block or chain branch may replace the block containing the transaction.

This replacement is known as a blockchain reorganization, or reorg.

A reorg can temporarily remove a transaction, change its position, or replace it with a conflicting transaction.

After strong finality, honest nodes should not accept a conflicting history without a major consensus failure.

Finality does not mean that reversal is physically impossible under every imaginable event.

It means that reversal would violate the protocol’s normal security assumptions and may require substantial malicious voting power, computational resources, destroyed stake, or exceptional community intervention.

What Is Time to Finality?

Time to finality is the period between submitting or including a transaction and reaching the blockchain’s defined final settlement state.

The measurement may begin when a user broadcasts a transaction, when a validator receives it, or when it enters a block.

These starting points can produce different reported results.

A protocol may advertise one-second finality while a user experiences a longer total delay because the transaction waited before block inclusion.

The complete user experience can include wallet signing, network transmission, mempool waiting, block production, consensus voting, and application indexing.

The official explanation of time to finality distinguishes settlement time from transaction throughput.

Accurate comparisons should identify both the starting event and the finality standard being measured.

How Does Fast Finality Work?

Fast finality normally depends on validators reaching agreement about a proposed block within a small number of communication rounds.

A block producer first gathers valid transactions and proposes a block.

Other validators verify the block’s transactions, signatures, state changes, and protocol rules.

The validators then send signed votes expressing support for the block or its chain.

When the required voting threshold is reached, the protocol records the block as committed, rooted, accepted, justified, or finalized.

The exact name depends on the consensus design.

Cryptographic signatures allow nodes and applications to verify that the required validators approved the block.

Fast networks attempt to complete this process with limited message delays while retaining acceptable security and decentralization.

Transaction Inclusion vs. Transaction Finality

Transaction inclusion means that a block producer placed a transaction inside a block.

Finality means that the block has received the protocol’s required settlement guarantee.

A transaction can be included without being final.

A wallet may display the transaction as successful as soon as it observes the proposed block.

A later fork or reorganization may still replace that block.

The risk is often small during normal operation, but it is not identical to full finality.

High-value applications may wait for a stronger commitment than ordinary wallet interfaces.

Fast Confirmation vs. Fast Finality

Fast confirmation is an early indication that a transaction is likely to remain in the accepted chain.

Fast finality is the stronger assurance that the transaction has passed the protocol’s final settlement threshold.

A confirmation may rely on one block producer, an observed vote, a fork-choice rule, or an application’s risk model.

Finality normally relies on a broader consensus condition.

Applications sometimes call an optimistic confirmation final even when the underlying chain still permits a reorganization.

Users should determine whether a displayed status means processed, confirmed, safe, rooted, committed, or finalized.

Fast Finality vs. Block Time

Block time is the average or scheduled interval between new blocks.

Time to finality is the amount of time required for a block to become sufficiently irreversible.

A blockchain can produce blocks every second but require many additional blocks before strong finality.

Another blockchain can have a longer block interval but finalize each committed block immediately after a supermajority vote.

A fast block time therefore does not prove that the network has fast finality.

Marketing materials that report only block time may leave out the period during which the block can still be reorganized.

Fast Finality vs. Transaction Throughput

Transaction throughput measures how many transactions a blockchain can process during a period.

It is commonly expressed as transactions per second.

Finality measures how quickly a completed transaction becomes settled.

A network can process many transactions while taking a relatively long time to finalize them.

A network can also finalize blocks quickly while supporting a smaller number of transactions.

Throughput and finality affect user experience, but they measure different properties.

Fast Finality vs. Low Latency

Latency is the delay between an action and an observed response.

A wallet may show a transaction within milliseconds because a server accepted and relayed it.

This low interface latency does not mean that validators finalized the transaction.

Network propagation, block inclusion, voting, and final settlement can continue after the application displays success.

Users should distinguish front-end response time from consensus finality.

Probabilistic Finality

Probabilistic finality means that the likelihood of a transaction being reversed decreases as more consensus work or voting support accumulates.

There may be no single protocol event that changes the block from unfinalized to permanently finalized.

Users instead choose a confirmation threshold based on the value and risk of the transaction.

Proof-of-work blockchains commonly use confirmation depth for this purpose.

Each additional block makes replacing the transaction’s history more difficult for an attacker.

The Bitcoin confirmation explanation states that every additional confirmation reduces the risk of reversal.

Probabilistic does not mean insecure.

It means the security level is expressed as a decreasing probability rather than a binary finality certificate.

Deterministic Finality

Deterministic finality generally means that a committed block cannot be replaced without violating the consensus protocol’s safety assumptions.

Byzantine fault-tolerant systems often provide this type of finality after a defined validator threshold signs the block.

The blockchain does not normally create two valid competing committed blocks at the same height while the assumed number of validators remains honest.

Users do not need to wait for a long chain of additional confirmations after the block is committed.

The guarantee still depends on assumptions about validator behavior, voting power, software correctness, and network conditions.

Economic Finality

Economic finality makes a conflicting finalized history possible only by placing substantial economic value at risk.

Proof-of-stake protocols can require validators to lock cryptocurrency as collateral.

Validators that sign prohibited conflicting votes may lose some or all of that stake through slashing.

Ethereum describes its finality as crypto-economic because reversing finalized history would require a consensus failure involving significant staked ETH.

The official Ethereum proof-of-stake documentation explains that checkpoint pairs require votes representing two-thirds of staked ETH to become finalized.

Byzantine Fault-Tolerant Finality

Byzantine fault tolerance allows a distributed system to reach agreement even when some participants fail or act maliciously.

Many BFT blockchains assume that less than one-third of voting power is Byzantine.

A typical process includes proposal, prevote, precommit, and commit stages.

A block becomes committed after more than two-thirds of voting power sends the required approval.

The CometBFT consensus specification documents a round containing proposal, prevote, precommit, and commit steps.

If the network does not receive enough votes, it starts another round rather than finalizing an uncertain block.

This behavior favors safety over immediate progress during severe communication problems.

Why Does the Two-Thirds Threshold Matter?

A two-thirds supermajority creates overlap between any two groups large enough to finalize conflicting blocks.

Two groups containing more than two-thirds of total voting power must share more than one-third of that power.

Validators in the overlap would need to support conflicting decisions.

The protocol can identify such behavior through signed votes and may punish it.

This intersection property is a foundation of many BFT and proof-of-stake finality designs.

The threshold is normally based on voting power or stake rather than a simple count of validator computers.

What Is a Finality Certificate?

A finality certificate is cryptographic evidence that the required validator threshold approved a block or checkpoint.

It may contain individual signatures, an aggregate signature, a quorum certificate, a commit certificate, or another proof of voting support.

Nodes can verify the certificate without trusting the application that supplied it.

Light clients and cross-chain systems may use finality certificates to confirm events without executing every consensus message themselves.

The certificate is only as secure as the validator set, signature scheme, consensus rules, and method used to identify the correct validator weights.

Fork Choice vs. Finality

A fork-choice rule selects the chain that nodes should follow when several possible branches exist.

A finality rule determines which older blocks should no longer be replaced.

A fork-choice decision near the chain head can change as new blocks and votes arrive.

A finalized checkpoint creates a boundary that honest nodes should not cross.

Some blockchain protocols combine a rapidly updating fork-choice chain with a slower finalized chain.

This allows applications to use recent blocks optimistically while waiting for stronger settlement.

Fast Finality in Ethereum

Ethereum produces blocks in 12-second slots but currently takes about 15 minutes to reach full protocol finality under normal conditions.

Validators vote on checkpoint relationships through Ethereum’s proof-of-stake consensus mechanism.

Blocks become finalized after the required checkpoint voting conditions are satisfied.

Recently proposed blocks can be considered increasingly safe before full finality, but they do not yet have the strongest protocol guarantee.

Ethereum is researching single-slot finality, which would allow a block to be proposed and finalized within the same slot.

The current Ethereum single-slot finality roadmap states that this design remains in the research phase and is not expected to activate immediately.

The research highlights a tradeoff between finality speed, validator participation, network communication, and hardware requirements.

Fast Finality in CometBFT-Style Consensus

CometBFT provides a round-based BFT consensus engine for deterministic blockchain applications.

A proposer broadcasts a candidate block for a particular height and round.

Validators prevote after checking the proposal.

They precommit when the required prevote conditions are satisfied.

A node enters the commit stage when it observes more than two-thirds of precommit voting power for the same block.

The committed block does not require a long sequence of later blocks to obtain the same protocol-level status.

Actual finality time depends on block interval settings, validator locations, network quality, application execution, and whether more than one round is needed.

Fast Finality in Avalanche Consensus

Avalanche-family consensus protocols use repeated random sampling instead of requiring every validator to communicate directly with every other validator in each step.

A validator repeatedly asks small random subsets of validators about their preferences.

Confidence grows when the sampled responses consistently support the same decision.

The protocol accepts the decision after its confidence threshold is reached.

The official Avalanche consensus documentation describes probabilistic safety guarantees with sub-second finality under normal conditions.

The probability of conflicting acceptance can be tuned through sampling and confidence parameters.

This design demonstrates that fast finality can be probabilistic while still offering a defined acceptance event.

Fast Finality in Solana

Solana exposes different commitment levels so applications can choose between speed and stronger settlement assurance.

A processed transaction has been handled by a node but may exist on a branch that does not survive.

A confirmed block has received a supermajority of ledger votes.

Solana defines finality as the state in which nodes representing two-thirds of stake have a common root.

The official Solana terminology distinguishes confirmation from finality.

Applications may use a faster commitment level for low-risk interactions and a finalized level for higher-value settlement.

The decision should account for the possibility that an early processed branch is abandoned.

Fast Finality and Proof of Work

Proof-of-work blockchains usually do not provide a BFT-style finality certificate for every transaction.

A block becomes harder to reverse as miners add more proof of work above it.

Users often wait for a chosen number of confirmations before considering a payment settled.

The correct waiting period depends on transaction value, attacker resources, network hash power, and the receiver’s risk tolerance.

A zero-confirmation transaction offers fast visibility but weak protection against conflicting spends.

A first confirmation provides stronger evidence, while additional confirmations further reduce reversal risk.

The Bitcoin developer guide explains how proof of work protects earlier transaction records from modification.

Why Fast Finality Matters for Crypto Payments

A merchant needs confidence that a customer cannot reverse a payment after receiving goods or services.

Long finality delays can make blockchain payments inconvenient at stores, online checkouts, transportation systems, and automated machines.

Fast finality allows the merchant to release a product or service sooner.

It can also reduce the amount of working capital trapped in unsettled transfers.

The merchant must still verify that the transaction paid the correct address, asset, network, and amount.

Finality does not protect against a worthless token, incorrect exchange rate, fraudulent product order, or compromised wallet.

Why Fast Finality Matters for Crypto Trading

Trading platforms and settlement systems move assets between wallets, collateral accounts, and smart contracts.

Slow finality can delay deposits, withdrawals, collateral recognition, and arbitrage activity.

Fast finality allows confirmed funds to become usable more quickly.

It can reduce exposure to a deposit disappearing through a chain reorganization.

Operational policies may still require additional review for security, compliance, wallet maintenance, or large transactions.

A blockchain reaching finality does not guarantee immediate credit in an account maintained by a separate service.

Why Fast Finality Matters for Decentralized Finance

Decentralized finance applications combine lending, trading, collateral, liquidation, and price calculations in automated smart contracts.

A reorganization can remove one transaction while leaving users or offchain systems acting on the earlier result.

Fast finality reduces the period during which recent state changes remain uncertain.

It can improve the safety of rapid collateral transfers, liquidations, and multi-step financial operations.

Smart contract bugs, oracle manipulation, insufficient collateral, and liquidity failure remain possible after finality.

Finality proves the permanence of accepted execution rather than the economic correctness of the application.

Why Fast Finality Matters for Blockchain Bridges

A cross-chain bridge observes an event on one blockchain and causes a corresponding event on another blockchain.

If the bridge acts before source-chain finality, the original event may disappear through a reorganization.

The destination-chain transfer may already have become final, leaving the bridge undercollateralized.

Waiting for source finality reduces this risk.

Fast finality allows a bridge to provide faster transfers without accepting as much reorganization exposure.

The bridge must also verify destination finality, validator signatures, smart contract security, message ordering, and replay protection.

Why Fast Finality Matters for Layer 2 Networks

A layer 2 network may provide a rapid local confirmation before the transaction reaches final settlement on its underlying blockchain.

The local confirmation can come from a sequencer, committee, or separate consensus protocol.

Stronger settlement may depend on publishing transaction data, submitting a proof, passing a challenge period, or reaching base-layer finality.

A user can therefore experience several different finality stages for the same transaction.

The fastest stage improves application responsiveness.

The later stage provides stronger protection against operator failure or invalid state transitions.

Applications should state which layer and guarantee their finality claim describes.

Fast Finality and Atomic Transactions

Atomic execution means that all operations inside one blockchain transaction succeed together or fail together.

Finality means that the accepted result becomes difficult or impossible to reverse under the protocol assumptions.

A transaction can be atomic but not yet final.

A finalized transaction can also contain an atomic failure that consumed fees but made no requested state change.

The two properties solve different problems.

Fast Finality and Double-Spend Protection

A double spend attempts to use the same cryptocurrency units in conflicting transactions.

Consensus determines which conflicting transaction becomes part of the accepted history.

Before finality, a receiver faces some risk that another branch will replace the observed payment.

Fast finality shortens this risk period.

It does not prevent a user from broadcasting conflicting transactions before either one is finalized.

The protection comes from the network settling on one accepted history.

Fast Finality and Network Partitions

A network partition occurs when groups of validators cannot communicate reliably with one another.

A safe BFT protocol may stop finalizing blocks when no connected group has the required supermajority.

This temporary loss of progress protects against two isolated groups finalizing conflicting histories.

A protocol that continues finalizing on both sides of a severe partition may weaken safety.

Fast finality claims normally assume that messages can reach enough honest validators within expected timing conditions.

Finality can become slower or stop entirely when those assumptions fail.

Safety vs. Liveness

Safety means that honest nodes do not finalize conflicting blockchain histories.

Liveness means that the blockchain continues processing and finalizing new transactions.

A consensus protocol can protect safety by refusing to finalize when it lacks sufficient votes.

This choice temporarily harms liveness.

Reducing voting thresholds to preserve progress can weaken safety.

Fast finality protocols must balance rapid decisions with resistance to network delays and malicious behavior.

Communication Overhead and Fast Finality

Validators must receive proposals, transaction data, votes, and finality evidence before they can make a safe decision.

Traditional BFT designs can require communication that grows rapidly as the validator set expands.

Signature aggregation, committees, gossip networks, and random sampling can reduce this burden.

Shorter finality targets give validators less time to exchange and verify messages.

This may require faster internet connections, more processing power, and highly available infrastructure.

Excessive requirements can reduce the number of people capable of operating validators.

Fast Finality and Decentralization

A small validator committee can communicate and reach agreement quickly.

It may also be easier for a limited group to coordinate, censor transactions, suffer correlated failure, or become compromised.

A very large and geographically distributed validator set can improve participation and resilience.

It also creates more signatures, messages, and network delays.

Fast finality is therefore partly an engineering tradeoff between latency, overhead, validator count, and fault tolerance.

A finality time should not be evaluated without examining who can validate and how voting power is distributed.

Can Fast Finality Fail?

Yes, finality can be delayed when too much validator power is offline, partitioned, or running faulty software.

A consensus bug can cause nodes to disagree about valid blocks or votes.

A coordinated attack may prevent the network from reaching its required threshold.

Validators may also finalize a harmful but technically valid transaction because consensus does not judge whether a user was scammed.

A severe safety failure can produce conflicting finalized histories.

Resolving that event may require software updates, governance decisions, slashing, or community coordination.

What Is a Finality Delay?

A finality delay occurs when the blockchain continues producing or observing blocks but does not advance its strongest finalized state as expected.

The delay can result from validator outages, network congestion, message loss, client bugs, malicious voting, or a partition.

Recently included transactions may remain usable by risk-tolerant applications.

High-value systems may pause deposits, withdrawals, bridge messages, or settlement until finality resumes.

A delay is not automatically proof that transactions have been reversed.

It means the protocol has not yet provided its normal finality guarantee.

Can a Finalized Transaction Be Reversed?

A finalized transaction should not be reversed during normal protocol operation.

Reversal may still be possible after a catastrophic consensus failure, a critical software defect, or exceptional social intervention.

Economic-finality systems may require dishonest validators to sacrifice substantial stake.

BFT systems may require more faulty voting power than their safety assumptions permit.

Probabilistic systems do not create the same binary finality boundary, although reversal can become extraordinarily unlikely.

The word final should always be understood within the stated consensus and governance assumptions.

How Can Users Check Transaction Finality?

Users can begin by checking the transaction through a reputable block explorer for the correct blockchain.

The explorer may display confirmations, commitment level, checkpoint status, root status, or a finalized label.

Users should verify whether the explorer is showing transaction inclusion or full finality.

High-value applications can query their own node rather than relying on one external interface.

Developers can compare the transaction’s block height with the latest finalized block or checkpoint.

Several independent data sources can reduce reliance on one delayed or inaccurate provider.

How Developers Should Handle Finality

Developers should identify every confirmation level exposed by the target blockchain.

Low-value interface updates may use an early confirmation for better responsiveness.

Asset withdrawals, bridge messages, and irreversible offchain actions should use a stronger threshold.

Applications should detect chain reorganizations and remove data derived from abandoned blocks.

They should also detect finality delays instead of assuming settlement always occurs on schedule.

Database records should retain the block hash and height associated with each onchain event.

Users should be told clearly when a transaction is pending, confirmed, safe, or finalized.

How to Compare Fast Finality Claims

Identify whether the reported time begins at transaction broadcast or block inclusion.

Determine whether the result is an optimistic confirmation, probabilistic acceptance, deterministic commit, or economic finality.

Check the validator threshold and assumptions about dishonest voting power.

Review whether the measurement represents ordinary conditions or a guaranteed upper bound.

Examine validator count, voting-power concentration, and hardware requirements.

Determine what happens during a network partition or validator outage.

Check whether the application relies on another blockchain for final settlement.

Review historical finality delays and consensus incidents where reliable data is available.

Common Misconceptions About Fast Finality

Fast block production does not automatically mean fast finality.

High transaction throughput does not automatically mean fast finality.

A wallet displaying success does not prove protocol-level settlement.

One confirmation does not have the same security meaning on every blockchain.

Probabilistic finality is not necessarily weak finality.

Deterministic finality still depends on consensus assumptions and correct software.

Fast finality does not prevent wallet theft, smart contract bugs, or fraud.

A layer 2 confirmation may not be the same as base-layer finality.

A bridge transaction may require finality on more than one blockchain.

A claimed one-second finality time does not mean every user transaction settles exactly one second after submission.

Frequently Asked Questions

What does fast finality mean in crypto?

Fast finality means that a blockchain can give a transaction a strong irreversibility guarantee within a relatively short period.

How fast is fast finality?

There is no universal threshold, although the term commonly describes settlement within seconds or a small number of blocks or voting rounds.

Is finality the same as confirmation?

No, confirmation can be an earlier and weaker status than the blockchain’s strongest finality state.

Is finality the same as block time?

No, block time measures block production frequency, while finality measures how long a block takes to become settled.

Is finality the same as throughput?

No, throughput measures transaction capacity, while finality measures settlement delay.

What is time to finality?

Time to finality is the period required for a transaction to reach the network’s defined irreversible or economically irreversible state.

What is probabilistic finality?

It is a model in which the probability of reversal decreases as more blocks, work, samples, or votes support the transaction.

What is deterministic finality?

It is a consensus guarantee that a committed block cannot be replaced while the protocol’s fault assumptions remain valid.

What is economic finality?

It is finality protected by economic penalties that make conflicting consensus votes extremely costly.

What is BFT finality?

It is finality produced by a Byzantine fault-tolerant protocol after the required validator supermajority approves a block.

Why do many protocols require two-thirds of stake?

The threshold ensures that two conflicting supermajorities must overlap by at least one-third of voting power.

What is a finality certificate?

It is cryptographic evidence showing that the required validator threshold approved a block or checkpoint.

Can a transaction be included but not finalized?

Yes, recently included transactions can remain exposed to chain reorganizations until stronger consensus support develops.

Can fast finality prevent double spending?

It shortens the period in which a conflicting spend may replace an observed transaction.

Can a finalized block be reorganized?

Not during normal operation, although a catastrophic consensus failure or exceptional intervention may challenge that guarantee.

Does Ethereum have fast finality?

Ethereum produces blocks quickly but currently takes about 15 minutes to reach full protocol finality.

Does Ethereum currently use single-slot finality?

No, single-slot finality remains a research-stage roadmap objective.

What is a Solana commitment level?

It is a setting that lets applications select a level of confirmation ranging from recent processing to stronger finalized state.

Does Avalanche use deterministic finality?

Its official consensus documentation describes tunable probabilistic safety with sub-second finality under normal conditions.

Does CometBFT provide immediate finality?

A CometBFT block is committed after more than two-thirds of voting power precommits to it, without requiring a long later confirmation chain.

Why can finality become delayed?

Finality can slow or stop because of validator outages, network partitions, software bugs, message delays, or malicious behavior.

Can a blockchain process transactions during a finality delay?

Some protocols can continue producing blocks even when their strongest finalized checkpoint is not advancing.

Does fast finality make a blockchain secure?

It improves settlement speed but does not by itself guarantee decentralization, smart contract safety, custody security, or resistance to every attack.

Why does fast finality matter for bridges?

A bridge needs confidence that a source-chain event will not disappear after the destination-chain action is executed.

Why does fast finality matter for payments?

It allows merchants to release goods or services sooner with less exposure to transaction reversal.

Is a layer 2 transaction final immediately?

It may receive a fast local confirmation while stronger settlement on the underlying blockchain occurs later.

How can I check whether a crypto transaction is final?

Check the network’s official commitment definition and compare the transaction block with the latest finalized block or checkpoint.

Can a block explorer show incorrect finality information?

Yes, an explorer may be delayed, misconfigured, or using a weaker confirmation status than expected.

Does faster finality require more powerful hardware?

It can because validators have less time to receive, aggregate, verify, and process consensus messages.

Can a smaller validator set finalize faster?

It often can communicate more efficiently, but reduced participation may create greater centralization and correlated-failure risks.

What should developers do before acting on a transaction?

They should choose a confirmation threshold appropriate to the transaction’s value, reversibility, and dependence on other systems.

Conclusion

Fast finality is the ability of a blockchain to make transactions effectively irreversible within a short and predictable period.

It is different from fast block time, low interface latency, and high transaction throughput.

A transaction can appear in a block or wallet before it reaches the network’s strongest settlement state.

Blockchain protocols provide finality through methods including confirmation depth, supermajority voting, slashing-backed checkpoints, BFT commit certificates, and repeated random sampling.

Each method creates different assumptions about validator honesty, network communication, economic penalties, and reversal probability.

Fast finality improves crypto payments, trading settlement, decentralized finance, cross-chain bridges, and layer 2 user experiences.

It also involves tradeoffs involving communication overhead, hardware requirements, validator-set size, decentralization, safety, and liveness.

Users should examine the exact meaning of processed, confirmed, safe, committed, rooted, or finalized on the blockchain they use.

Developers should match their required confirmation level to the value and irreversibility of the action triggered by the transaction.

A credible fast-finality claim explains the consensus threshold, security assumptions, starting point, expected network conditions, and behavior during failures rather than presenting only one speed number.

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