
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
In the regular pyramid \( S M N P Q \) ( \( S \) is the apex), points \( H \) and \( F \) are the midpoints of edges \( M N \) and \( N P \) respectively, point \( E \) lies on the segment \( S H \) such that \( S H = 3 \) and \( S E = \frac{9}{1} \). The distance from point \( S \) to the line \( E F \) is \( \sqrt{5} \). Find the volume of the pyramid. Additionally, given a sphere of radius 1 with center at point \( S \), all possible regular tetrahedra \( A B C D \) are considered such that points \( C \) and \( D \) lie on the line \( E F \), and the line segment \( A B \) touches the sphere at one of the points on the segment \( A B \). Find the minimum edge length of the considered tetrahedra.

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