
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
Let $p$: For any $x \in \mathbb{R}$, it holds that $x^2 - 2x > a$, and $q$: There exists $x_0 \in \mathbb{R}$, such that $x_0^2 + 2ax_0 + 2 - a = 0$. If the proposition $p \vee q$ is true, and the proposition $p \wedge q$ is false, find the range of the real number $a$.

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