
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 dihedral angle $\alpha-l-\beta$ is $60^{\circ}$, points $P$ and $Q$ are on planes $\alpha$ and $\beta$ respectively. The distance from $P$ to plane $\beta$ is $\sqrt{3}$, and the distance from $Q$ to plane $\alpha$ is $2 \sqrt{3}$. What is the minimum distance between points $P$ and $Q$?

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