
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
Points  $C\neq D$  lie on the same side of line  $AB$  so that  $\triangle ABC$  and  $\triangle BAD$  are congruent with  $AB = 9$ ,  $BC=AD=10$ , and  $CA=DB=17$ .  The intersection of these two triangular regions has area  $\tfrac mn$ , where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n$ .

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