
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]


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Given the binomial expansion ${\left(2x+ \frac{1}{ \sqrt{x}}\right)}^{n}\left(n∈{N}^{\*}\right)$, where $N^*$ represents the set of positive integers, the ratio of the binomial coefficients of the second term to the third term is $2:5$. Complete the following tasks:

(I) Find the value of $n$;

(II) Find the term containing ${x}^{3}$ in the expansion;

(III) Calculate the value of the expression ${2}^{6}C\_{6}^{0}+{2}^{5}C\_{6}^{1}+{2}^{4}C\_{6}^{2}+{2}^{3}C\_{6}^{3}+{2}^{2}C\_{6}^{4}+2C\_{6}^{5}$.

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