
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
A regular tetrahedron $ABCD$ has its centroid marked as point $P$. Compute the ratio $\frac{s}{S}$ where:
- $s$ is the sum of the distances from $P$ to the planes containing $DAB$, $DBC$, and $DCA$.
- $S$ is the sum of the distances from $P$ to the lines containing $AB$, $BC$, and $CA$.
A) $\frac{\sqrt{2}}{3}$
B) $1$
C) $\frac{\sqrt{2}}{2}$
D) $\sqrt{2}$
E) $2$

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