
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 $f(x)=|x-a|+2x$.
$(1)$ When $a=3$, solve the inequality $f(x) \geqslant 3$;
$(2)$ If the solution set of the inequality $f(x) \leqslant 0$ with respect to $x$ is $\{x|x \leqslant -2\}$, find the value of the real number $a$.

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