
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 $(1-\frac{1}{2}x)^n = a\_0 + a\_1x + a\_2x^2 + a\_3x^3 + \ldots + a\_nx^n$, if $|a\_0|$, $|a\_1|$, $|a\_2|$ form an arithmetic sequence.

1. Find the middle term in the expansion of $(1-\frac{1}{2}x)^n$.
2. Find the sum of the coefficients of all odd powers of $x$ in the expansion of $(1-\frac{1}{2}x)^n$.
3. Find the value of $a\_1 + 2a\_2 + 3a\_3 + \ldots + na\_n$.

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