
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
If $C_{n}^{0} + \frac{1}{2}C_{n}^{1} + \frac{1}{3}C_{n}^{2} + \ldots + \frac{1}{n+1}C_{n}^{n} = \frac{31}{n+1}$, then the term with the largest coefficient in the expansion of $(1+x)^{2n}$ is \_\_\_\_\_\_.

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