
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
Calculate the range of the real number $a$ given the following conditions: Proposition $p$: For all $x \in [1,2]$, $x^2 - a \geq 0$. Proposition $q$: There exists an $x_0 \in \mathbb{R}$, such that $x + (a-1)x_0 + 1 < 0$. "Proposition $p$ or proposition $q$" is true, while "proposition $p$ and proposition $q$" is false.

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