
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
Triangle  $ABC$  has a right angle at  $C$ ,  $AC = 3$  and  $BC = 4$ .  Triangle  $ABD$  has a right angle at  $A$  and  $AD = 12$ .  Points  $C$  and  $D$  are on opposite sides of  $\overline{AB}$ .  The line through  $D$  parallel to  $\overline{AC}$  meets  $\overline{CB}$  extended at  $E$ .  If  $\frac{DE}{DB} = \frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, then find the value of $m + n$.

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