
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 that $a \in \mathbb{R}$, proposition $p$: "$\forall x \in [1, 2], x^2 - a \leqslant 0$", and proposition $q$: "$\exists x_0 \in \mathbb{R}, x_0^2 + 2ax_0 + 2 - a = 0$".

(1) If proposition $p$ is true, find the range of values for the real number $a$.

(2) If the proposition "$p \vee q$" is true and the proposition "$p \wedge q$" 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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