
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 the tetrahedron with a volume of $V$, and the area of its $i$-th face denoted as $S_i (i=1,2,3,4)$, and for any point $Q$ inside this tetrahedron, the distance to the $i$-th face is $H_i (i=1,2,3,4)$, if $\frac{S_1}{1}=\frac{S_2}{2}=\frac{S_3}{3}=\frac{S_4}{4}=k$, determine the value of $H_1+2H_2+3H_3+4H_4$.

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