
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 \(a\) be a fixed real number. Define \(M(t)\) to be the maximum value of \(-x^{2} + 2ax + a\) over all real numbers \(x\) with \(x \leq t\). Determine a polynomial expression in terms of \(a\) that is equal to \(M(a-1) + M(a+2)\) for every real number \(a\).

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