
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 numbers \([a], [2a], \ldots, [Na]\) are all distinct from each other, and the numbers \([1/a], [2/a], \ldots, [M/a]\) are also distinct from each other. Find all such \(a\).

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