
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
On side $AC$ of triangle $ABC$, a circle is constructed such that $AC$ acts as its diameter. This circle intersects sides $AB$ and $BC$ at points $D$ and $E$, respectively. The angle $\angle EDC$ is $30^\circ$, $EC=1$, and the area of triangle $DBE$ is half the area of triangle $ABC$. Find the length of segment $BO$, where $O$ is the intersection point of segments $AE$ and $CD$.

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