
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 $a\in\mathbb{R}$, consider the following propositions:
- Proposition $P$: "For all $x\in[1,2], x^{2}-a\geq 0$",
- Proposition $Q$: "There exists $x\in\mathbb{R}$ such that $x^{2}+2ax+2-a=0$".

$(1)$ If proposition $P$ is true, find the range of values for the real number $a$;
$(2)$ If the proposition $P \vee Q$ (OR) is true, but proposition $P \wedge Q$ (AND) is false, find the range of values for the real number $a$.

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