
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
A regular octahedron \( ABCDEF \) is given such that \( AD, BE \), and \( CF \) are perpendicular. Let \( G, H \), and \( I \) lie on edges \( AB, BC \), and \( CA \) respectively such that \(\frac{AG}{GB} = \frac{BH}{HC} = \frac{CI}{IA} = \rho\). For some choice of \(\rho > 1\), \( GH, HI \), and \( IG \) are three edges of a regular icosahedron, eight of whose faces are inscribed in the faces of the octahedron. Find \(\rho\).

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