
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
The distance from a point in space to a plane is defined as follows: draw a perpendicular line from a point in space to the plane, and the distance between this point and the foot of the perpendicular is called the distance from this point to the plane. Given planes $\alpha$, $\beta$, and $\gamma$ are mutually perpendicular to each other, point $A$ is on plane $\alpha$, and the distance from point $A$ to planes $\beta$ and $\gamma$ is 3. Point $P$ is a moving point on plane $\alpha$, satisfying that the distance from $P$ to plane $\beta$ is twice the distance from $P$ to point $A$. The minimum value of the distance from points on the trajectory of point $P$ to plane $\gamma$ is \_\_\_\_\_\_.

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