
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 $n$ is a positive integer, the coefficient of the $x^3$ term in the expansions of $(1+x)^{2n}$ and $(1+2x^{3})^{n}$ are the same. Find:
1. The value of $n$.
2. The term with the maximum binomial coefficient in the expansion of $(1+2x^{3})^{n}$.

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