
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 $f(x)=|x+a|$ ($a\in\mathbb{R}$).  
$(1)$ If the solution set of $f(x)\geqslant |2x+3|$ is $[-3,-1]$, find the value of $a$;  
$(2)$ If for all $x\in\mathbb{R}$, the inequality $f(x)+|x-a|\geqslant a^{2}-2a$ always holds, find the range of the real number $a$.

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