
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 \( X = \{0, a, b, c\} \) and \( M(X) = \{ f \mid f: X \rightarrow X \} \) be the set of all functions from \( X \) to itself. Define the addition operation \( \oplus \) on \( X \) as given in the following table:
\[
\begin{array}{|c|c|c|c|c|}
\hline
\oplus & 0 & a & b & c \\
\hline
0 & 0 & a & b & c \\
\hline
a & a & 0 & c & b \\
\hline
b & b & c & 0 & a \\
\hline
c & c & b & a & 0 \\
\hline
\end{array}
\]

1. Determine the number of elements in the set:
\[
S = \{ f \in M(X) \mid f((x \oplus y) \oplus x) = (f(x) \oplus f(y)) \oplus f(x), \forall x, y \in X \}.
\]

2. Determine the number of elements in the set:
\[
I = \{ f \in M(X) \mid f(x \oplus x) = f(x) \oplus f(x), \forall x \in X \}.
\]

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