
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 propositions p and q, where:
- p: For all $x$ in the interval $[1, 2]$, $x^2 - a \geq 0$.
- q: There exists some $x_0 \in \mathbb{R}$ such that $x_0^2 + (a-1)x_0 + 1 < 0$.

If the compound statement "p or q" is true and "p and q" is false, determine the range of possible values for the real number $a$.

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