
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 $k$ be any one of the numbers 1, 2, 3, 4, or 5. Determine the necessary and sufficient condition for each value of $k$ separately, such that there exists a tetrahedron with $k$ edges each of length $a$ units and the remaining $(6-k)$ edges each of length 1 unit.

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