
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
For constants  $a,\ b,\ c,\ d$  consider a process such that the point  $(p,\ q)$  is mapped onto  the point  $(ap+bq,\ cp+dq)$ .
Note :  $(a,\ b,\ c,\ d)\neq (1,\ 0,\ 0,\ 1)$ . Let  $k$  be non-zero constant. All points of the parabola  $C: y=x^2-x+k$  are mapped onto  $C$  by the process.

(1) Find  $a,\ b,\ c,\ d$ .

(2) Let  $A'$  be the image of the point  $A$  by the process. Find all values of  $k$  and the coordinates of  $A$  such that the tangent line of  $C$  at  $A$  and the tangent line of  $C$  at  $A'$  formed by the process are perpendicular at the origin.

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