
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
Determine all positive integers \( M \) for which the sequence \( a_{0}, a_{1}, a_{2}, \ldots \), defined by \( a_{0} = \frac{2M+1}{2} \) and \( a_{k+1} = a_{k} - \left\lfloor a_{k} \right\rfloor \) for \( k = 0, 1, 2, \ldots \), contains at least one integer term.

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