
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 the expansion of the function $f(x) = (1 + x)^m + (1 + x)^n$ where $m, n \in \mathbb{N}$ and the coefficient of $x$ is $19$.

1. Find the minimum value of the coefficient of the $x^2$ term in the expansion of $f(x)$.
2. When the coefficient of the $x^2$ term is at its minimum, find the coefficient of the $x^7$ term in the expansion of $f(x)$.

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