
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
Let points  $A,B,$  and  $C$  lie on a line such that  $AB=1, BC=1,$  and  $AC=2.$  Let  $C_1$  be the circle centered at  $A$  passing through  $B,$  and let  $C_2$  be the circle centered at  $A$  passing through  $C.$  Find the area of the region outside  $C_1,$  but inside  $C_2.$ 

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