
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 $f\left(x\right)=|mx-1|-|x+2a^{2}|$.  
$(1)$ When $a=1$ and $m=2$, find the set $M$ of values of $x$ for which $f\left(x\right) \gt 2$;  
$(2)$ When $m=1$, if the inequality $f\left(x\right) \lt -3a$ holds for any real number $x$, find the range of real numbers for $a$.

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