
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
Please choose one of the following three conditions and fill in the blank line below, then solve the problem.<br/>① The ratio of the coefficient of the 5th term to the coefficient of the 3rd term is $56:3$;<br/>② The sum of the binomial coefficients of the 2nd term and the 3rd term from the end is $55$;<br/>③ ${C}_{n+1}^{2}-{C}_{n}^{n-2}=10$.<br/>Given the expansion of ${(\frac{\sqrt{x}}{2}-\frac{1}{\sqrt[3]{x}})}^{n}$,____.<br/>$(1)$ Find the term with the largest binomial coefficient in the expansion;<br/>$(2)$ Find the term containing $x^{5}$ in the expansion.

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