
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 [1,2]\\), \\(x^2 - a \geqslant 0\\), and proposition \\(Q\\) be: the equation \\(\frac{x^2}{a+1} + \frac{y^2}{a-2} = 1\\) represents a hyperbola.

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

\\((2)\\) If the proposition \\(P \vee Q\\) is true, and the proposition \\(P \wedge Q\\) is false, find the range of the real number \\(a\\).

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