
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
C is a circle of unit radius. For any line L, define d(L) to be the distance between the two points of intersection if L meets C in two points, and zero otherwise. For any point P, define f(P) to be the maximum value of d(L) + d(L') for two perpendicular lines through P. Which P have f(P) > 2?

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