
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
Given triangle $ABC$ with $m \angle A = 45^\circ$, $m \angle B = 75^\circ$, and $AC = 16$ units. If $\overline{BD} \perp \overline{AC}$ and $\overline{CE} \perp \overline{AB}$, find the length of segment $EC$. Assume $m \angle DBC = 2m \angle ECB$. 

[asy]
draw((0,0)--(8,0)--(5,5)--cycle);
draw((5,0)--(5,5));
draw((8,0)--(4,2.5));
label("$A$",(0,0),W);
label("$B$",(5,5),N);
label("$C$",(8,0),E);
label("$D$",(5,0),S);
label("$E$",(4,2.5),NW);
[/asy]

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