
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
Given a quadrilateral with an area of $S$ and its $i$-th side length denoted as $a\_i (i = 1, 2, 3, 4)$, let $P$ be a point inside the quadrilateral with a distance of $h\_i$ to the $i$-th side. If $\frac{a\_1}{1} = \frac{a\_2}{2} = \frac{a\_3}{3} = \frac{a\_4}{4} = k$, then $h\_1 + 2h\_2 + 3h\_3 + 4h\_4 = \frac{2S}{k}$. By analogy, given a triangular pyramid with a volume of $V$ and its $i$-th face area denoted as $S\_i (i = 1, 2, 3, 4)$, let $Q$ be a point inside the triangular pyramid with a distance of $d\_i$ to the $i$-th face. If $\frac{S\_1}{1} = \frac{S\_2}{2} = \frac{S\_3}{3} = \frac{S\_4}{4} = k$, then $d\_1 + 2d\_2 + 3d\_3 + 4d\_4$ equals $\_\_\_\_\_\_\_\_$.

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