# Difficult Geometry - Circle #1

Geometry Level pending

Circles $$O_1$$ and $$O_2$$ intersect with each other at points $$\text{A}$$ and $$\text{B}$$.

$$O_1$$ and $$O_2$$ has a radius of $$3$$ and $$5$$, and has a center of $${\text{C}}_1$$ and $${\text{C}}_2$$, respectively.

Point $$\text{D}$$ internally divides $$\overline{\text{AB}}$$ with a ratio of $$1:3$$.

$$\overline{\text{EF}}$$ and $$\overline{\text{KP}}$$ intersect at point $$\text{D}$$.

$$\overline{\text{EF}}$$ is a chord of $$O_1$$, with point $$\text{F}$$ being inside circle $$O_1$$ and $$\text{E}$$ being outside it, satisfying $$\overline{\text{AE}}<\overline{\text{BE}}$$ and $$\overline{\text{AF}}>\overline{\text{BF}}$$.

Point $$\text{K}$$ is on circle $$O_2$$ and inside circle $$O_1$$. Also, it satisfies $$\angle\text{EFK}=20^{\circ}$$, $$\angle\text{EKF}=130^{\circ}$$ and $$\angle{\text{C}}_1\text{FK}<\angle{\text{C}}_1\text{FE}$$.

Point $$\text{P}$$ satisfies $$\angle\text{EPF}=50^{\circ}$$.

Find the value of $$\overline{{\text{C}}_2 \text{P}}$$.

If you think the given information is too less to figure out the answer, submit $$-1$$ as your answer. And if you think the given information doesn't make sense, submit $$0$$ as your answer.

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