
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
Let  $a_1$ ,  $a_2$ ,  $a_3$ ,  $a_4$ , and  $a_5$  be real numbers satisfying
\begin{align*}
a_1a_2 + a_2a_3 + a_3a_4 + a_4a_5 + a_5a_1 & = 20,
a_1a_3 + a_2a_4 + a_3a_5 + a_4a_1 + a_5a_2 & = 22.
\end{align*}Then the smallest possible value of  $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2$  can be expressed as  $m + \sqrt{n}$ , where  $m$  and  $n$  are positive integers. Compute  $100m + n$ .

*Proposed by Ankan Bhattacharya*

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