
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 $(1- \frac{1}{2}x)^n = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots + a_nx^n$, if $|a_0|$, $|a_1|$, $|a_2|$ form an arithmetic sequence.  
(1) Find the middle term of the expansion of $(1- \frac{1}{2}x)^n$.  
(2) Find the sum of the coefficients of the terms with 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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