
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 tetrahedron $T$ is intersected by different planes $p_1, p_2, ..., p_k$. Let $S$ be the total surface area of $T$ and let $P = \bigcup_{j=1}^k p_j$. The intersection of $P$ and $S$ comprises the union of all line segments that connect the midpoints of each pair of edges belonging to the same face of $T$. Find the difference between the maximum and minimum possible values of $k$ required to form all such segments.
A) 2
B) 4
C) 6
D) 8
E) 10

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