
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
Suppose we have an infinite cone \(\mathcal{C}\) with apex \(A\) and a plane \(\pi\). The intersection of \(\pi\) and \(\mathcal{C}\) is an ellipse \(\mathcal{E}\) with major axis \(BC\), such that \(B\) is closer to \(A\) than \(C\), and \(BC = 4\), \(AC = 5\), \(AB = 3\). Suppose we inscribe a sphere in each part of \(\mathcal{C}\) cut by \(\mathcal{E}\) with both spheres tangent to \(\mathcal{E}\). What is the ratio of the radii of the spheres (smaller to larger)?

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