
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 variable tangent  $t$  to the circle  $C_1$ , of radius  $r_1$ , intersects the circle  $C_2$ , of radius  $r_2$  in  $A$  and  $B$ . The tangents to  $C_2$  through  $A$  and  $B$  intersect in  $P$ .
Find, as a function of  $r_1$  and  $r_2$ , the distance between the centers of  $C_1$  and  $C_2$  such that the locus of  $P$  when  $t$  varies is contained in an equilateral hyperbola.**Note**: A hyperbola is said to be *equilateral* if its asymptotes are perpendicular.

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