
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 ${(2{x}^{3}+\frac{1}{{x}^{2}})}^{n}(n∈{N}^{*})$.  
$(1)$ If the binomial coefficients of the 5th and 6th terms in its expansion are equal, find $n$;  
$(2)$ If there exists a constant term in the expansion, find the minimum value of $n$.

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