
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 a quadratic function $f(x)=ax^{2}+bx+c$.
(1) If $a=c > 0$, $f(1)=1$, find the expression for $g(a)$, which is the sum of the maximum and minimum values of $f(x)$ for any $x∈[-2,2]$;
(2) If $a$, $b$, $c$ are positive integers, and the function $f(x)$ has two distinct zeros in the interval $(-\frac{1}{4}, \frac{1}{4})$, find the minimum value of $a+b+c$.

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