
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
What is the shortest distance between the plane \(Ax + By + Cz + 1 = 0\) and the ellipsoid \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)? You may find it convenient to use the notation \(h = (A^2 + B^2 + C^2)^{-1/2}\), \(m = (a^2A^2 + b^2B^2 + c^2C^2)^{1/2}\). What is the algebraic condition for the plane not to intersect the ellipsoid?

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