
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
Find $CY$ in the diagram if $BY$ bisects $\angle ABC$ and $D$ is the midpoint of $AC$.

[asy]
import markers;

real t=.56;
pair A=(0,0);
pair B=(2,2);
pair C=(3,0);
pair Y=t*B+(1-t)*C;
pair D=(0.5*A + 0.5*C);

draw(A--B--C--cycle);
draw(B--Y);
draw(A--D);

label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$Y$",Y,SE);
label("$D$",D,S);

label("$36$", B--Y, NE);
label("$15$", B--C, S);
label("$20$", A--C, S);

[/asy]

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