
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 $p$ be $\forall x \in [-2,-1], x^2 - a \geq 0$, and proposition $q$ be $\exists x \in \mathbb{R}, x^2 + 2ax - (a-2) = 0$.  
$(1)$ If proposition $p$ is true, determine the range of real number $a$.  
$(2)$ If proposition "$p \lor q$" is true and proposition "$p \land q$" is false, determine the range of real number $a$.

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