
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 ratio of the area of $\triangle BDY$ to the area of $\triangle ADY$ in the diagram if $DY$ bisects $\angle ADB$. [asy]
import markers;

real t=35/(35+40);
pair A=(-17.57,0);
pair B=(11.43,0);
pair D=(0,30);
pair Y=t*A+(1-t)*B;

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

label("$A$",A,SW);
label("$B$",B,E);
label("$D$",D,N);
label("$Y$",Y,NE);

label("$28$", .5*(B+A), S);
label("$35$", .5*(B+D), NE);
label("$40$", .5*(A+D), NW);
[/asy]

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