
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, mayoral elections are conducted as follows. If in a particular round of voting no candidate receives more than half of the votes, another round is held with all candidates except the one with the least number of votes. (No two candidates ever have an equal number of votes; if a candidate receives more than half the votes, they are elected mayor and the election ends.) In each round, each voter votes for one candidate. If their candidate reaches the next round, the voter votes for them again. If their candidate is eliminated, all their voters vote for a single remaining candidate.

In the latest election, 2002 candidates were running. Ostap Bender became the mayor, having taken the $k$-th place in terms of the number of votes in the first round. Determine the highest 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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