
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 $60\%$ of voters would not like to see any of the candidates as their deputy, but under equal conditions, they are ready to sell their vote to the candidate who offers them more money.

It is known that if any candidate offers 1 monetary unit for a vote, he will gain only one additional supporter, if he offers 2 monetary units - two additional supporters, and so on. The voting in country Y is open, so only after the voting will the candidate know exactly who voted for him, and he will give the money to voters immediately after the 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 the ballots, i.e., each of the twenty-five voters must vote either for Ratibor, or for Nikifor, or against all, and the candidates themselves are not allowed to vote. In addition, Ratibor (and Nikifor) can compete unfairly, but he is not rich, so the maximum amount of money he is willing to give to buy votes is such that it would be enough to win the election if the competitor behaves honestly. Nikifor knows Ratibor's condition and his possible expenses, so he is always ready to pay more for any voter's bribery than Ratibor would offer. To win the election, any candidate needs to gain at least $50\%$ of the votes. Monetary units in this country can only take integer values.

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

(b) Find the minimal expenses for Nikifor for bribing voters to win the election, which he would be willing to bear if he knew for sure that Ratibor would not compete honestly, and even if after the voting neither candidate can determine to which group each voter belonged.

(c) Nikifor found a way to reliably find out before the election the minimum amount of money each voter is willing to take for selling their vote, and now the candidate will be able to individually make deals with all the voters he needs first. It is known that voters keep their word: if after the agreement with Nikifor Ratibor approaches them, none of them will change their decision to give their vote for Nikifor, no matter how much Ratibor offers them for their vote. The cost of information about an individual voter is $\mathrm{F}$ monetary units. What is the maximum amount $\mathrm{F}$ 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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