
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
Solve the following problems:  
$(1)$ If $\left(x-3\right)\left(x-4\right)=x^{2}+mx+n$, then the value of $m$ is ______, and the value of $n$ is ______;  
$(2)$ If $(x+a)(x+b)=x^2-3x+\frac{1}{3}$,  
① Find the value of $\left(a-1\right)\left(b-1\right)$;  
② Find the value of $\frac{1}{a^2}+\frac{1}{b^2}$.

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