
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
(Full Score 14 points) Given a quadratic function $f(x)$ whose leading coefficient is $a$, and the solution set to the inequality $f(x) > -2x$ is $(1,3)$.

(1) If the equation $f(x) + 6a = 0$ has two equal roots, find the explicit formula for $f(x)$.

(2) If the maximum value of $f(x)$ is positive, determine the range of the real number $a$.

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