
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
Find the smallest constant M, so that for any real numbers  $a_1, a_2, \dots a_{2023} \in [4, 6]$  and  $b_1, b_2, \dots b_{2023} \in [9, 12] $  following inequality holds:  $$  \sqrt{a_1^2 + a_2^2 + \dots + a_{2023}^2} \cdot  \sqrt{b_1^2 + b_2^2 + \dots + b_{2023}^2} \leq M \cdot \left ( a_1 b_1 + a_2 b_2 + \dots  + a_{2023} b_{2023} \right)  $$  *Proposed by Zaza Meliqidze, Georgia*

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