
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
In a three-dimensional space, we have three mutually perpendicular planes: $\alpha$, $\beta$, and $r$. Let there be a point $A$ on plane $\alpha$. Point $A$ is at a distance of $1$ from both planes $\beta$ and $r$. Let $P$ be a variable point on plane $\alpha$ such that the distance from $P$ to plane $\beta$ is $\sqrt{2}$ times the distance from $P$ to point $A$. The question asks for the minimum distance from points on the trajectory of $P$ to plane $r$.

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