
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
It is known that in the Cartesian coordinate system \(xOy\), line \(l\) passing through the point \((1,0)\) is perpendicular to the line \(x-y+1=0\), and \(l\) intersects the circle \(C: x^2+y^2=-2y+3\) at points \(A\) and \(B\). The area of triangle \(\triangle OAB\) is \_\_\_\_\_\_.

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