
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
Define  $\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2$  for every two points  $A = (x_A, y_A)$  and  $B = (x_B, y_B)$  in the plane.
Let  $S$  be the set of points  $(x,y)$  in the plane for which  $x,y \in \left\{ 0,1,\dots,100 \right\}$ .
Find the number of functions  $f : S \to S$  such that  $\left\lVert A-B \right\rVert \equiv \left\lVert f(A)-f(B) \right\rVert \pmod{101}$  for any  $A, B \in S$ .

*Proposed by Victor Wang*

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