
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 $a\in \mathbb{R}$, let proposition $p$: "$\forall x\in [1,2], x^2-a\geqslant 0$", and proposition $q$: "$\exists x\in \mathbb{R}, x^2+2ax+2=0$".

$(1)$ If proposition $p$ is true, find the range of real number $a$.

$(2)$ If the proposition "$p\vee q$" is true, and the proposition "$p\wedge q$" is false, find the range of real number $a$.

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