
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
In the diagram below  $ \angle CAB, \angle CBD$ , and  $\angle CDE$  are all right angles with side lengths  $AC = 3$ ,  $BC = 5$ ,  $BD = 12$ , and  $DE = 84$ . The distance from point  $E$  to the line  $AB$  can be expressed as the ratio of two relatively prime positive integers,  $m$  and  $n$ . Find  $m + n$ .

[asy]
size(300);
defaultpen(linewidth(0.8));
draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8));
label(" $A$ ",origin,SW);
label(" $B$ ",(0,4),dir(160));
label(" $C$ ",(3,0),S);
label(" $D$ ",(6,8),dir(80));
label(" $E$ ",(30,-4),E);[/asy]

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