
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 $f(x) = ax^2 + bx + c$, where $a < 0$, the solution set of the inequality $f(x) > -2x$ is $(1,3)$.  
(1) If the equation $f(x) + 6a = 0$ has two equal real roots, find the expression for $f(x)$.  
(2) If the maximum value of $f(x)$ is positive, find the range of values for the real number $a$.

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