
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
The triangle ABC has sides AB = 137, AC = 241, and BC =200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. Determine the length of CD.
[asy]

pair A = (2,6);
pair B = (0,0);
pair C = (10,0);
pair D = (3.5,0) ;
pair E = (3.1,2);
draw(A--B);
draw(B--C);
draw(C--A);
draw (A--D);
dot ((3.1,1.7));
label  ("E", E, dir(45));
label ("A", A, dir(45));
label ("B", B, dir(45));
label ("C", C, dir(45));
label ("D", D, dir(45));
draw(circle((1.8,1.3),1.3)); 
draw(circle((4.9,1.7),1.75)); 
[/asy]

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