
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
Planes $\alpha$, $\beta$, and $\gamma$ are mutually perpendicular. Point $A$ is on plane $\alpha$, and the distance from point $A$ to planes $\beta$ and $\gamma$ is both 3. $P$ is a moving point on plane $\alpha$, and the distance from $P$ to plane $\beta$ is twice the distance from $P$ to point $A$. Find the minimum distance from points on the trajectory of $P$ to plane $\gamma$.

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