
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  $r$  and  $s$  be positive real numbers, and let  $A=(0,0)$ ,  $B=(1,0)$ ,  $C=(r,s)$ , and  $D=(r+1,s)$  be points on the coordinate plane. Find a point  $P$ , whose coordinates are expressed in terms of  $r$  and  $s$ , with the following property: if  $E$  is any point on the interior of line segment  $AB$ , and  $F$  is the unique point other than  $E$  that lies on the circumcircles of triangles  $BCE$  and  $ADE$ , then  $P$  lies on line  $\overleftrightarrow{EF}$ .

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