
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 $f(x)=x^{2}+2ax-a+2$.

$(1)$ If for any $x \in \mathbb{R}$, $f(x) \geqslant 0$ always holds, find the range of the real number $a$;

$(2)$ If for any $x \in [-1,1]$, $f(x) \geqslant 0$ always holds, find the range of the real number $a$;

$(3)$ If for any $a \in [-1,1]$, $x^{2}+2ax-a+2 > 0$ always holds, find the range of the real number $x$.

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