
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
Inside the tetrahedron \(A B C D\), points \(X\) and \(Y\) are given. The distances from point \(X\) to the faces \(A B C\), \(A B D\), \(A C D\), \(B C D\) are \(14, 11, 29, 8\) respectively. The distances from point \(Y\) to the faces \(A B C\), \(A B D\), \(A C D\), \(B C D\) are \(15, 13, 25, 11\) respectively. Find the radius of the inscribed sphere of the tetrahedron \(A B C D\).

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