
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 sequence is defined by $u_{0}=M+\frac{1}{2}$ and $u_{n+1}=u_{n}\left\lfloor u_{n}\right\rfloor$. Determine all integers $M \geq 1$ for which the sequence takes on an integer value.

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