
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 the coefficient of $x$ in the expansion of the function $f(x) = (1 + x)^m + (1 + 2x)^n$ ($m, n \in \mathbb{N}^*$) is $11$, find the sum of the coefficients of the odd power terms of $x$ in the expansion of $f(x)$ when the coefficient of $x^2$ is at its minimum value.

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