
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 the sum of the binomial coefficients in the expansion of ${{\left( \sqrt{x}+\dfrac{1}{\sqrt[3]{x}} \right)}^{n}}$ is 64, find the coefficient of the $x^2$ term in $1+(1-x)^2+(1-x)^3+…+(1-x)^n$;

(2) Given that the binomial coefficients of the 5th, 6th, and 7th terms in the expansion of ${{\left( \dfrac{1}{2}+2x \right)}^{n}}$ form an arithmetic sequence, find the coefficient of the binomial term with the largest coefficient in the expansion.

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