
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
(1) Given that in the expansion of $(1+2x)^n$, the coefficients of the 6th and 7th terms are equal, find $n$ and the term with the maximum binomial coefficient.  
(2) Given $$(2- \sqrt {3}x)^{50}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{50}x^{50}$$, find the value of $$(a_{0}+a_{2}+a_{4}+\ldots+a_{50})^{2}-(a_{1}+a_{3}+a_{5}+\ldots+a_{49})^{2}$$.

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