
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 the function $f(x) = x - 1 - a\ln x$.

$(1)$ If $f(x) \geqslant 0$, find the value of $a$;

$(2)$ Let $m$ be an integer, and for any positive integer $n$, $\left( 1+\frac{1}{2} \right)\left( 1+\frac{1}{{2}^{2}} \right)\cdots \left( 1+\frac{1}{{2}^{n}} \right) < m$, find the minimum value of $m$.

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