
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 that \( a_{1}, a_{2}, \cdots, a_{n} \) are all negative real numbers and satisfy \( a_{1} + a_{2} + \cdots + a_{n} = 1 \), find the maximum value of \( m \) and the minimum value of \( M \) such that the inequality \( m \leqslant \sum_{1}^{n} \sqrt{1 + 3a_{i}} \leqslant M \) always holds.

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