
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 in the expansion of $\left(3x-1\right)^{n}$, the binomial coefficients of the $4$th and $6$th terms are equal.<br/>$(1)$ Find the coefficient of the $x^{2}$ term;<br/>$(2)$ If ${(3x-1)}^{n}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+⋯+{a}_{n}{x}^{n}$, find the value of ${2}^{n}{a}_{0}+{2}^{n-1}{a}_{1}+⋯+2{a}_{n-1}+{a}_{n}$.

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