
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 the propositions: 
p: For all $x$ in the interval $[1,2]$, $x^2 - a^2 \geq 0$. 
q: There exists an $x$ in $\mathbb{R}$ such that $x^2 + (a-1)x + 1 < 0$. 
If either p or q is true, but not both, determine the range of the real number $a$.

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