
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 and let $E$ be the centroid of face $ABC$. Denote by $t$ the sum of the distances from $E$ to the planes $DAB$, $DBC$, and $DCA$, and by $T$ the sum of the distances from $E$ to the vertex $D$ and to the lines containing $AB$, $BC$, and $CA$. Calculate the ratio $\frac{t}{T}$.
A) $\frac{\sqrt{2}}{2}$
B) $\frac{2\sqrt{2}}{3}$
C) $\frac{3\sqrt{2}}{5}$
D) $\frac{\sqrt{6}}{2}$
E) $\frac{\sqrt{3}}{2}$

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