
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 three logicians are to distribute 10 gold coins. Confident in their logical abilities, they decide to distribute the coins according to the following scheme:

(1) They draw lots to determine their order: 1, 2, 3;

(2) Logician 1 proposes a distribution plan, which is then voted on by the remaining members. If the plan receives approval from at least half of the remaining members (including the proposer), the coins are distributed according to the plan. Otherwise, Logician 1 receives only 2 coins and is eliminated from further distribution and voting;

(3) Logician 2 proposes a plan, and the remaining members vote. The plan is implemented only if it receives approval from at least half of the remaining members (including the proposer). Otherwise, Logician 2 also receives 2 coins and is eliminated;

(4) The remaining coins are all given to Logician 3.

Each logician is capable of rigorous logical reasoning and can rationally judge their own gain. For Logician 1 to maximize their gain, the number of coins they would propose for Logicians 1, 2, and 3, respectively, is \_\_\_\_\_\_\_\_\_\_.

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