
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
 $\vartriangle ABC$  has side lengths  $AB = 4$  and  $AC = 9$ . Angle bisector  $AD$  bisects angle  $A$  and intersects  $BC$  at  $D$ . Let  $k$  be the ratio  $\frac{BD}{AB}$  . Given that the length  $AD$  is an integer, find the sum of all possible  $k^2$ .

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