
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
A circle has its center at $(1,1)$ and passes through the origin. Determine its equation.

A) ${{\left( x-1 \right)}^{2}}+{{\left( y-1 \right)}^{2}}=1$
B) ${{\left( x+1 \right)}^{2}}+{{\left( y+1 \right)}^{2}}=1$
C) ${{\left( x+1 \right)}^{2}}+{{\left( y+1 \right)}^{2}}=2$
D) ${{\left( x-1 \right)}^{2}}+{{\left( y-1 \right)}^{2}}=2$

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