
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  $A = (0,0)$ ,  $B=(-1,-1)$ ,  $C=(x,y)$ , and  $D=(x+1,y)$ , where  $x > y$  are positive integers. Suppose points  $A$ ,  $B$ ,  $C$ ,  $D$  lie on a circle with radius  $r$ .  Denote by  $r_1$  and  $r_2$  the smallest and second smallest possible values of  $r$ .  Compute  $r_1^2 + r_2^2$ .

*Proposed by Lewis Chen*

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