
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 an isosceles acute triangle, the angle at the base is \(\alpha\), and the area is \(S\). Find the area of the triangle whose vertices are the feet of the altitudes of the given triangle.

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