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

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