
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, $f(x)=|x+2a|$.

$(1)$ If the solution set of $f(x) < 4-2a$ is $\{x|-4 < x < 4\}$, find the value of the real number $a$;

$(2)$ If for $a=1$, $f(x)-f(-2x) \leqslant x+m$ holds for any real number $x$, find the range of values for the real number $m$.

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