
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
Find the number of functions \( f \) from the set \( S = \{ 0, 1, 2, \ldots, 2020 \} \) to itself such that, for all \( a, b, c \in S \), all three of the following conditions are satisfied:
(i) If \( f(a) = a \), then \( a = 0 \);
(ii) If \( f(a) = f(b) \), then \( a = b \); and
(iii) If \( c \equiv a + b \pmod{2021} \), then \( f(c) \equiv f(a) + f(b) \pmod{2021} \).

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