
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 $m, n \in \mathbb{N}$, and $f(x) = (1+x)^m + (1+x)^n$.
1. When $m=n=7$, $f(x) = a_7x^7 + a_6x^6 + \ldots + a_1x + a_0$, find $a_0 + a_2 + a_4 + a_6$.
2. If the coefficient of the expanded form of $f(x)$ is 19 when $m, n$ vary, find the minimum value of the coefficient of $x^2$.

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