
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
Let  $ABC$  be a right triangle with  $\angle ACB = 90^{\circ}$  and centroid  $G$ . The circumcircle  $k_1$  of triangle  $AGC$  and the circumcircle  $k_2$  of triangle  $BGC$  intersect  $AB$  at  $P$  and  $Q$ , respectively. The perpendiculars from  $P$  and  $Q$  respectively to  $AC$  and  $BC$  intersect  $k_1$  and  $k_2$  at  $X$  and  $Y$ . Determine the value of  $\frac{CX \cdot CY}{AB^2}$ .

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