
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 \( a_{i} \in (0,1] (1 \leqslant i \leqslant 45) \),

find the maximum value of \( \lambda \) such that the inequality
$$
\sqrt{\frac{45}{\sum_{i=1}^{45} a_{i}}} \geqslant 1 + \lambda \prod_{i=1}^{45}\left(1 - a_{i}\right)
$$
holds.

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