
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 three distinct lines be given by:
$$
\begin{array}{l}
l_{1}: a x + 2b y + 3(a+b+1) = 0, \\
l_{2}: b x + 2(a+b+1) y + 3a = 0, \\
l_{3}: (a+b+1) x + 2a y + 3b = 0,
\end{array}
$$

Find the necessary and sufficient condition for them to intersect at a common point.

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