
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 $f(x)$ with a negative leading coefficient $a$ ($a < 0$), and 1 and 3 being the two roots of the function $y = f(x) + 2x$. If the equation $f(x) + 6a = 0$ has two equal roots, find the explicit formula for $f(x)$.

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