
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
Let $f(x) = (1+x)^m + (1+x)^n$ where the coefficient of $x$ in the expansion is 19, and $m, n \in \mathbb{N}^*$.  
(1) Find the minimum value of the coefficient of $x^2$ in the expansion of $f(x)$.  
(2) For $m$ and $n$ that minimize the coefficient of $x^2$ in the expansion of $f(x)$, find the coefficient of $x^7$ in the expansion.

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