
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
Given that rectangle \(ABCD\) folds upward along the line segment \(EF\) which is parallel to \(AD\) (with point \(E\) moving along segment \(AB\) and point \(F\) moving along segment \(CD\)), resulting in a triangular prism \(ABECDF\), where \(AB = 5\) and \(AC = \sqrt{34}\),

1. If \(\triangle ABG\) is a right triangle with \(G\) being a point on segment \(EF\), determine the range of lengths of segment \(EG\).
2. If the length of \(EG\) is the largest integer within the determined range and \(AB\) is minimized, find the value of the dihedral angle \(C-E F-D\).
3. Under the conditions in (1) and (2), find the volume of the tetrahedron \(A-BFG\).

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