
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 line intersects the circles $k_{1}$ and $k_{2}$, which do not share a common point, cutting chords $A B$ and $C D$ of equal length, respectively. The points of intersection are $A, B, C,$ and $D$. A point $P$ is such that $P A$ is tangent to circle $k_{1}$ and $P D$ is tangent to circle $k_{2}$. Determine the ratio $\frac{P A}{P D}$ using the radii of the two circles.

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