
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 right triangle \(ABC\), angle \(ACB\) is a right angle. Let \(E\) be the point where the angle bisector of \(\angle ABC\) intersects side \(AC\). Point \(D\) is the midpoint of side \(AB\), and \(O\) is the point of intersection of segments \(BE\) and \(CD\). A perpendicular line is drawn from point \(O\) to \(BO\) meeting side \(BC\) at point \(F\). It is given that \(FC = b\) and \(OC = \frac{3b}{2}\). Find the area of triangle \(ABC\).

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