
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 polyhedron has  $12$  faces and is such that:***(i)*** all faces are isosceles triangles,***(ii)*** all edges have length either  $x$  or  $y$ ,***(iii)*** at each vertex either  $3$  or  $6$  edges meet, and***(iv)*** all dihedral angles are equal.

Find the ratio  $x/y.$ 

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