
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
If $a^{m}=a^{n}$ ($a \gt 0$ and $a\neq 1$, $m$ and $n$ are positive integers), then $m=n$. Use the above conclusion to solve the following problems:<br/>
$(1)$ If $2\div 8^{x}\cdot 16^{x}=2^{5}$, find the value of $x$;<br/>
$(2)$ If $2^{x+2}+2^{x+1}=24$, find the value of $x$;

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