
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
$(1)$ In the expansion of the binomial ${({5x-\frac{1}{{\sqrt[3]{x}}}})^n}$, the sum of all binomial coefficients is $64$. Find the coefficient of the term containing $x^{2}$ in the expansion.<br/>$(2)$ Given $\left(1-x\right)\left(2x+1\right)^{3}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots +a_{6}x^{6}$. Find ① $a_{2}+a_{4}+a_{6}$; ② the value of $a_{3}$.

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