
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
Let $f\left(x\right)=(1+x^{2})^{m}-\left(1+x\right)^{2n}$, where $m\in \mathbb{N}^{*}$, $n\in \mathbb{N}^{*}$.<br/>$(1)$ When $m=4$ and $n=3$, let the coefficient of $x^{i}$ in the expansion of $f\left(x\right)$ be $a_{i}$ for $i=0,1,2,3,4,5,6,8$. Find the value of $a_{3}+a_{4}$;<br/>$(2)$ If the coefficient of $x^{2}$ in the expansion of $f\left(x\right)$ is $20$, find the minimum value of $\frac{m}{n}$.

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