
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 \(ABCD\) has side lengths \(AB = 6\), \(BD = 6\sqrt{2}\), \(BC = 10\), \(AC = 8\), \(CD = 10\), and \(AD = 6\). The distance from vertex \(A\) to face \(BCD\) can be written as \(\frac{a \sqrt{b}}{c}\), where \(a, b, c\) are positive integers, \(b\) is square-free, and \(\operatorname{gcd}(a, c) = 1\). Find \(100a + 10b + c\).

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