
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 $(1+ \frac{1}{2}x)^{m}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+…+a_{m}x^{m}$, where $a_{0}$, $a_{1}$, $a_{2}$ form an arithmetic sequence.

1. Find the middle term in the expanded form of $(1+ \frac{1}{2}x)^{m}$;
2. Find the sum of all coefficients of odd powers of $x$ in the expanded form of $(1+ \frac{1}{2}x)^{m}$;
3. Find the term with the maximum coefficient in the expanded form of $(1+ \frac{1}{2}x)^{m+6}$.

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