
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 that the function $y= \sqrt{x+2} \cdot \sqrt{5-x}$ has a domain defined as set $Q$, and set $P=\{x|a+1 \leq x \leq 2a+1\}$.
(1) If $a=3$, determine the intersection of the complement of $P$ in the real numbers with $Q$;
(2) If $P \subseteq Q$, find the range of the real number $a$.

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