
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 the function $f\left(x\right)=|x+a|+|x+3a|$.<br/>$(1)$ When $a=-1$, find the solution set of the inequality $f\left(x\right) \lt 4$;<br/>$(2)$ If the minimum value of $f\left(x\right)$ is $2$, and $\left(a-m\right)\left(a+m\right)=\frac{4}{{n}^{2}}$, find the minimum value of $\frac{1}{{m}^{2}}+n^{2}$.

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