
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
Find the equation of the circle whose center lies on the line $y=x$ in the first quadrant, passes through the point $(-1,2)$, and whose chord intercepted by the $x$-axis has a length of $4\sqrt{2}$. (The given answer is ${{\left( x-3 \right)}^{2}}+{{\left( y-3 \right)}^{2}}=17$)

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