
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 the smallest value of $n$ ($n \in \mathbb{N}^*$) that makes the expansion of the binomial ${{(3x+\frac{1}{x\sqrt{x}})}^{n}}$ contain a constant term is $a$, then $\int_{0}^{a}{({x}^{2}+2x-1)}dx=\_\_\_\_\_\_\_\_\_\_$.

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