
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the city of Udoev, the mayoral election is conducted as follows. If in a given round of voting, none of the candidates receive more than half of the votes, the next round is held with all candidates except the one with the least number of votes. (No two candidates ever receive an equal number of votes; if a candidate secures more than half the votes, they become the mayor, and the election ends.) In each round, each voter votes for one candidate. If this candidate advances to the next round, the voter votes for them again in the next round. If the candidate is eliminated, all of that candidate's voters vote for the same remaining candidate.

In the current election, 2002 candidates are running. Ostap Bender became the mayor, having placed $k$-th in the first round by the number of votes. Determine the largest possible value of $k$ if Ostap was elected a) in the 1002nd round; b)* in the 1001st round.

Present the answer in LaTex format: \boxed{Your answer}
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