
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 the length of segment \(AY\) in the diagram, where \(CY\) bisects \(\angle ACB\).

[asy]
import markers;

real t=.4;
pair A=(0,0);
pair B=(4,3);
pair C=(1,2);
pair Y=t*A+(1-t)*B;

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

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

label("$35$",.5*(B+Y),SE);
label("$42$",.5*(B+C),N);
label("$28$",.5*(A+C),NW);

[/asy]

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