
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

The remaining 40% of voters do not want to see any of the candidates as their deputy, but under equal conditions, they are willing to vote for the candidate who offers them the most money.

It is known that if a candidate offers 1 monetary unit per vote, they will gain exactly one additional supporter; if they offer 2 monetary units, they will gain two additional supporters, and so on. Voting in country Y is open, therefore only after the voting, a candidate will know exactly who voted for them and will give the voters the money immediately after their expression of will. It is also known that there are only two candidates in the election, that all voters are required to come to the polls and will not spoil their ballots, i.e., each of the thirty-five voters will definitely vote either for Ratibor, for Nikifor, or against all, and the candidates themselves do not have the right to vote. Additionally, Ratibor (as well as Nikifor) can compete dishonestly, but he is not wealthy, so the maximum amount of money he is willing to pay to bribe voters is such that it would be enough to win the election if the competitor behaves honestly. Nikifor knows about Ratibor’s financial capabilities, so he is ready to always pay more for any voter's bribe than what Ratibor would offer. To win the election, any candidate needs to get at least 50% of the votes. Monetary units in this country can only take integer values.

(a) Write down the function of the vote offer for any candidate.

(b) Find the minimum expenses Nikifor would incur to bribe voters to win the election if he knew for sure that Ratibor would not compete honestly, and even after the voting, neither candidate could determine which group each voter belonged to.

(c) Nikifor found a way to reliably find out before the elections the minimum amount of money each voter is willing to accept for selling their vote, and now the candidate can first make agreements with all the necessary voters individually. It is known that voters keep their word: once they agree with Nikifor, even if Ratibor approaches them with a higher offer, they will not change their decision to vote for Nikifor. The cost of information about an individual voter is F monetary units. What is the maximum amount of F that Nikifor would be willing to pay for such information to win the election?

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