
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
For two circles, one can inscribe a quadrilateral within the first circle and circumscribe it around the second circle. Let \( r_1 \) and \( r_2 \) be the radii of the circles, and \( d \) be the distance between the centers of the circles. Then 
$$
\frac{1}{(r_1 - d)^2} + \frac{1}{(r_1 + d)^2} = \frac{1}{r_2^2}.
$$

Moreover, the converse of this proposition is also true. For circles with radii \( R \) and \( r \), respectively, and center distance \( d \), a quadrilateral can be inscribed in one circle and circumscribed around the other if and only if 
$$
\frac{1}{(R - d)^2} + \frac{1}{(R + d)^2} = \frac{1}{r^2}.
$$

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