
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
A group of ten treasure hunters found a chest of gold coins with 100 coins that had been lost for more than two hundred years.

To facilitate organization, each hunter is assigned a number from 1 to 10 according to their hierarchy within the group. That is, hunter number 10 is the leader while number 1 cannot give orders to any of the others. They decided to use a certain form of "democracy" to divide the gold coins. Hunter 10 makes a proposal for the division of all the coins among the 10 hunters. Each hunter votes for or against the proposal. If half or more of the hunters vote in favor, this division is carried out. Otherwise, hunter 10 loses their turn and is excluded from the division. Hunter 9 then gets to make a proposal for dividing the 100 coins among hunters 1 through 9. Again, each hunter from 1 to 9 votes for or against, and if half or more agree, the division is made. Otherwise, hunter 9 loses their turn and gets no coins. The process continues, passing to hunter 8, and so on.

The hunters know that each coin cannot be split, as it is worth much more as a whole. Additionally, each hunter wants to gain as many coins as possible.

(a) Suppose the process has reached hunter 3. What proposal should he make to obtain the maximum gain while ensuring his proposal is accepted by hunters 1, 2, and 3?
(b) Suppose the process has reached hunter 4. The hunters are very clever and know the answer to the previous question. What proposal should hunter 4 make to achieve the maximum gain while ensuring it is accepted?

(c) Going back to the beginning of the problem and remembering that all the hunters are very clever, what proposal should hunter 10 make to achieve the maximum gain while ensuring his proposal is accepted in the vote?

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