
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
Line \( l_{1} \) passes through point \( P(3,2) \) and has an inclination angle of \( \arctan \frac{3}{4} \). If \( l_{1} \) intersects line \( l_{2}: x-2y+11=0 \) at point \( Q \), find the distance \( |PQ| \).

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