
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 the quadratic function $f(x) = x^2 + ax + \frac{a^2}{4} + 1$ ($a \in \mathbb{R}$), find the expression for $g(a)$, which represents the minimum value of the function $f(x)$ on the interval $[-1, 1]$.

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