
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 the sum of the coefficients of the first $3$ terms in the expansion of $({\frac{m}{{{x^2}}}+\sqrt{x}})^n$ (where $m$ is a positive real number) is equal to $37$.
$(1)$ Find the value of $n$;
$(2)$ If the coefficient of the term containing $\frac{1}{x}$ in the expansion is $112$, find the value of $m$.

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