
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 two lines $l\_1$: $ax + 2y + b = 0$ and $l\_2$: $(a - 1)x + y + b = 0$. If $l\_1$ is parallel to $l\_2$, and the distance between $l\_1$ and $l\_2$ is $\frac{\sqrt{2}}{2}$, find the value of $a \cdot b$.

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