
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 $f(x)=|x^{2}-2x+a-1|-a^{2}-2a$.  
$(1)$ When $a=3$, find the solution set for $f(x)\geqslant -10$;  
$(2)$ If $f(x)\geqslant 0$ holds for all $x\in\mathbb{R}$, find the range of values for $a$.

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