
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  $k$  be a positive real number. In the   $X-Y$  coordinate plane, let  $S$  be the set of all points of the form  $(x,x^2+k)$  where  $x\in\mathbb{R}$ . Let  $C$  be the set of all circles whose center lies in  $S$ , and which are tangent to  $X$ -axis. Find the minimum value of  $k$  such that any two circles in  $C$  have at least one point of intersection.

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