
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 that $m \angle A = 45^\circ$, $BC = 16$ units, $\overline{BD} \perp \overline{AC}$, $\overline{CE} \perp \overline{AB}$, and $m \angle DBC = 2m \angle ECB$, calculate the length of segment $EC$ which can be expressed in the form $a(\sqrt{b}+\sqrt{c})$ units where $b$ and $c$ have no perfect-square factors. What is the value of $a+b+c$?

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