
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 $({2x-\frac{1}{{\sqrt{x}}}})^5$.  
$(1)$ Find the coefficient of the term containing $\frac{1}{x}$ in the expansion.  
$(2)$ Let the sum of the binomial coefficients of the first three terms in the expansion of $({2x-\frac{1}{{\sqrt{x}}}})^5$ be $M$, and the sum of all coefficients in the expansion of $\left(1+ax\right)^{6}$ be $N$. If $4M=N$, find the value of the real number $a$.

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