
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
Given a positive integer \( n \). Find the largest integer \( M \) such that for any positive integers \( a_1, a_2, \ldots, a_n \), the following holds:
$$
\left\lfloor \sqrt{a_1} \right\rfloor + \left\lfloor \sqrt{a_2} \right\rfloor + \cdots + \left\lfloor \sqrt{a_n} \right\rfloor
\geq \left\lfloor \sqrt{a_1 + a_2 + \cdots + a_n + \operatorname{Min} \{a_1, a_2, \cdots, a_n\}} \right\rfloor,
$$
where \(\lfloor x \rfloor\) represents the greatest integer less than or equal to the real number \( x \).

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