
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
Let the solution set of the inequality $1-\frac{{a+1}}{{x+1}}＜0$ with respect to $x$ be $P$, and the solution set of the inequality $|x+2| \lt 3$ be $Q$.<br/>$(1)$ If $a=3$, find $P$;<br/>$(2)$ If $P\cup Q=Q$, find the range of positive number $a$.

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