
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
Two circles centered at points $A$ and $B$ each have a radius of 3. Point $O$ is the midpoint of $\overline{AB}$, and $OA=3\sqrt{3}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively. $E$ and $F$ are points on circles centered at $A$ and $B$ such that $\overline{EF}$ is a common tangent, and $\angle CAE = 60^\circ$ and $\angle DBF = 60^\circ$. Calculate the area of the shaded region $ECODF$.

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