
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 $\triangle ABC$, $\angle ABC = 60^\circ$, $O$ and $H$ are the circumcenter and orthocenter of $\triangle ABC$ respectively. Points $D$ and $E$ lie on sides $BC$ and $AB$ respectively, such that $BD = BH$, $BE = BO$, and $BO = a$. Find the area of $\triangle BDE$.

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