
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 three different lines
$$
\begin{array}{l}
l_{1}: ax + 2by + 3(a + b + 1) = 0, \\
l_{2}: bx + 2(a + b + 1)y + 3a = 0, \\
l_{3}: (a + b + 1)x + 2ay + 3b = 0.
\end{array}
$$

the necessary and sufficient condition for them to intersect at a single point is

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