
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
Proposition $p$: "For all $x$ in $[1,2]$, $x^2-a\geqslant 0$," Proposition $q$: "There exists $x_0$ in $\mathbb{R}$, such that $x_0^2+2ax_0+2-a=0$," If "Proposition $p$ AND Proposition $q$" is a false statement, find the range of the real number $a$.

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