
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 $m+n=6$ and $mn=-3$.  
$(1)$ When $a=2$, find the value of $a^{m}\cdot a^{n}-(a^{m})^{n}$;  
$(2)$ Find the value of $\left(m-n\right)^{2}+\left(m-4\right)\left(n-4\right)$.

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