
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 are willing to sell their vote to the candidate who offers them the most money.

It is known that if a candidate offers 1 unit of money per vote, they will gain only one additional supporter. If they offer 2 units of money per vote, they will gain two additional supporters, and so on. Voting in country Y is open, so a candidate will only know who voted for them after the voting concludes, and will distribute the money to the voters immediately after the election. It is also known that there are only two candidates in the election, that all voters must come to vote and cannot spoil their ballots, meaning that all thirty-five voters will definitely vote either for Ratibor, Nikiphor, or against all, and the candidates themselves do not have the right to vote. Furthermore, Ratibor (just like Nikiphor) can compete unfairly, but is not wealthy, so the maximum amount of money he is ready to give to bribe voters is just enough to win the election if his competitor behaves honestly. Nikiphor is aware of Ratibor's financial situation, so he is always ready to pay more than Ratibor for any vote. To win the election, a candidate needs to secure 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 Nikiphor would need to incur to bribe voters to win the election if he knows for certain that Ratibor will not compete fairly, and even after the election, no candidate can determine which group any voter belonged to.

(c) Nikiphor found a way to reliably learn before the election the minimum amount of money each voter is willing to accept for selling their vote, and now the candidate can individually negotiate with all the voters he needs first. It is known that voters keep their word: if they agree with Nikiphor, they will not change their decision to vote for him regardless of how much Ratibor offers them subsequently. The cost of information about each individual voter is \( \mathrm{F} \) monetary units. What is the maximum amount \( \mathrm{F} \) that Nikiphor would be willing to pay for such information in order to win the election?

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