
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 $a\neq 1$, $m$, $n$ are positive integers), then $m=n$. Can you use the above conclusion to solve the following problems? Give it a try, I believe you can do it!  
$(1)$ If $2\times 8^{x}\times 16^{x}=2^{22}$, find the value of $x$;  
$(2)$ If $(27^{-x})^{2}=(\frac{1}{9})^{-6}$, find the value of $x$;  
$(3)$ Given $p=5^{7}$, $q=7^{5}$, express $35^{35}$ in terms of $p$ and $q$.

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