
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 tetrahedron \( ABCD \) with \( AD = BC = 26 \), \( AC = BD = 42 \), and \( AB = CD = 50 \). For any point \( Y \) in space, define \( g(Y) = AY + BY + CY + DY \). Determine the least possible value of \( g(Y) \), which can be expressed as \( p\sqrt{q} \), where \( p \) and \( q \) are positive integers with \( q \) not divisible by the square of any prime. Find \( p+q \).

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