
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
Tetrahedron $EFGH$ has $EH=FG=20$, $EG=FH=26$, and $EF=GH=30$. For any point $Y$ in space, suppose $g(Y) = EY + FY + GY + HY$. Determine the least possible value of $g(Y)$ which can be expressed as $p\sqrt{q}$, where $p$ and $q$ are positive integers and $q$ is not divisible by the square of any prime. Find $p+q$.

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