
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 proposition $p: \forall x \in R$, $4mx^{2}+x+m\leqslant 0$.  
$(1)$ If $p$ is a true proposition, find the range of real number $m$;  
$(2)$ Proposition $q: \exists x \in \left[2,8\right]$, such that $m\log _{2}x\geqslant 1$. When $\neg p \wedge \neg q$ is a false proposition and $\neg q$ is a true proposition, find the range of real number $m$.

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