
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 that $m$ and $n$ are integers, solve the following problems:  
$(1)$ If $a \gt 0$, and $a^{m}=2$, $a^{n}=3$, find the value of $a^{m+n}$.  
$(2)$ If $x \gt 0$, and $x^{2n}=7$, find the value of $(x^{3n})^{2}$.

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