
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
On a circle of center  $O$ , let  $A$  and  $B$  be points on the circle such that   $\angle AOB = 120^o$ . Point  $C$  lies on  the small arc  $AB$  and point  $D$  lies on the segment  $AB$ . Let also  $AD = 2, BD = 1$  and  $CD = \sqrt2$ . Calculate the area of triangle  $ABC$ .

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