
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 relationship between line $l_{1}$: $ax+3y+6=0$, line $l_{2}$: $2x+(a+1)y+6=0$, and circle $C$: $x^{2}+y^{2}+2x=b^{2}-1(b > 0)$ is "parallel and intersecting", determine the range of $b$.

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