
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 $$(1+ \frac {1}{2}x)^{m}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots+a_{m}x^{m}$$, if $a_0$, $a_1$, $a_2$ form an arithmetic sequence.  
(1) Find the middle term of the expansion of $$(1+ \frac {1}{2}x)^{m}$$;  
(2) Find the sum of the coefficients of the terms with odd powers of $x$ in the expansion of $$(1+ \frac {1}{2}x)^{m}$$;  
(3) Find the term with the largest coefficient in the expansion of $$(1+ \frac {1}{2}x)^{m+6}$$.

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