
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 right triangular prism \( A_{1} B_{1} C_{1}-A B C \), \(\angle B A C=\frac{\pi}{2}\), \( A B = A C = A A_{1} = 1\). Let \( G \) and \( E \) be the midpoints of \( A_{1} B_{1} \) and \( C C_{1} \) respectively. Let \( D \) and \( F \) be points on the segments \( A C \) and \( A B \) respectively (excluding the endpoints). If \( G D \perp E F \), find the range of lengths of the segment \( D F\).

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