
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 real numbers $a$, $b$, $c$ are not all zero and form an arithmetic sequence, point $A(1, 2)$ is projected onto the moving line $l: ax + by + c = 0$ at point $P$, and point $Q$ lies on the line $3x - 4y + 12 = 0$. The minimum length of line segment $PQ$ is $\_\_\_\_\_\_$.

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