
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  $ABCD$  be a regular tetrahedron with side length  $1$ . Let  $EF GH$  be another regular tetrahedron such that the volume of  $EF GH$  is  $\tfrac{1}{8}\text{-th}$  the volume of  $ABCD$ . The height of  $EF GH$  (the minimum distance from any of the vertices to its opposing face) can be written as  $\sqrt{\tfrac{a}{b}}$ , where  $a$  and  $b$  are positive coprime integers. What is  $a + b$ ?


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