
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 convex polyhedron $Q$ has vertices $V_1, V_2, \ldots, V_n$, and originally $150$ edges. This polyhedron is intersected by planes $P_1, P_2, \ldots, P_n$, where each plane $P_k$ intersects every edge that meets at vertex $V_k$. Crucially, no two of these planes overlap within the volume or on the surface of $Q$. Each cut by a plane results in every involved edge being divided into three segments. Determine the total number of edges in the newly formed polyhedron $R$.
A) 300
B) 350
C) 400
D) 450
E) 500

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