
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 that $m, n \in \mathbb{N}$ and $f(x) = (1 + x)^m + (1 + x)^n$.

1. When $m = n = 7$, $f(x) = a_7x^7 + a_6x^6 + ... + a_1x + a_0$, find the value of $a_0 + a_2 + a_4 + a_6$.
2. If the coefficient of $x$ in the expansion of $f(x)$ is $19$, find the minimum value of the coefficient of $x^2$ as $m$ and $n$ vary.

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