
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
Let $ABCD$ be a regular tetrahedron. Let $P$ be a point inside the tetrahedron such that it is equidistant from all four vertices. Determine the ratio, $\frac{t}{T}$, where:
- $t$ is the sum of the distances from $P$ to the four vertices $A$, $B$, $C$, and $D$.
- $T$ is the sum of the distances from $P$ to the six edges $AB$, $BC$, $CA$, $AD$, $BD$, and $CD$.
A) $\frac{1}{3}$
B) $\frac{\sqrt{2}}{2}$
C) $\frac{2\sqrt{2}}{3}$
D) $\frac{3}{2}$

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