
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 elections proceed as follows. If in a given round of voting, no candidate receives more than half of the votes, the next round is held with all candidates except the one with the fewest votes. (No two candidates ever receive the same number of votes; if a candidate receives more than half the votes, they are elected mayor and the elections end.) Each voter votes for one candidate in each round. If the candidate progresses to the next round, the voter votes for them again. If the candidate is eliminated, all their voters will vote for a single remaining candidate.

In the latest elections, there were 2002 candidates. Ostap Bender was elected mayor, having placed $k$-th in the first round in terms of votes. Determine the largest possible value of $k$ if Ostap Bender 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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