
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
In the expansion of \((x+\sqrt{x}+1)^{2n+1} \ (n \in \mathbb{N}^*)\), what is the sum of the coefficients of the terms where \(x\) is raised to an integer power?

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