
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 set $A=\{x|x^{2}+3x-4\geqslant 0\}$, set $B=\left\{{x|{\frac{{x-2}}{x}≤0}}\right\}$.<br/>$(1)$ If $C=\{x\left|\right.2a \lt x \lt 1+a\}$, and $C\subseteq \left(A\cap B\right)$, find the range of real number $a$.<br/>$(2)D=\left\{{x|{{x^2}-({2m+\frac{1}{2}})x+m({m+\frac{1}{2}})≤0}}\right\}$, if $x\in A\cap B$ is a necessary but not sufficient condition for $x\in D$, determine whether real number $m$ exists, and if it does, find the range of $m$.

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