
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 the function $f(x, n) = (1+x)^n$, where $n \in \mathbb{N}^*$.
1. Find the term with the largest coefficient in the expansion of $f(x, 6)$.
2. If $f(i, n) = 32i$ (where $i$ is the imaginary unit), find the value of $C_n^1 - C_n^3 + C_n^5 - C_n^7 + C_n^9$.

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