
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 \(ABC\), the acute angles \(\alpha\) and \(\gamma\) (\(\alpha > \gamma\)) are given at the base \(AC\). From vertex \(B\), the altitude \(BD\) and the median \(BE\) have been drawn.

Find the area of triangle \(BDE\), if the area of triangle \(ACB\) is equal to \(S\).

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