
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
While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw 3 squares \(\square_{1}, \square_{2}, \square_{3}\) on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set \(\{1,2,3,4,5\}\) such that no integer is used more than once. Ana's goal is to minimize the minimum value \(M\) that the polynomial \(a_{1} x^{2} + a_{2} x + a_{3}\) attains over all real \(x\), where \(a_{1}, a_{2}, a_{3}\) are the integers written in \(\square_{1}, \square_{2}, \square_{3}\) respectively. Banana aims to maximize \(M\). Assuming both play optimally, compute the final value of \(100 a_{1} + 10 a_{2} + a_{3}\).

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