
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 the set \( S = \{A_0, A_1, A_2, A_3\} \), and define the operation " \(\oplus\)" on the set \( S \) as follows: \( A_i \oplus A_j = A_k \), where \( k \) is the remainder when \( i + j \) is divided by 4, \( i, j \in \{0, 1, 2, 3\} \). Determine the number of \( x \) (where \( x \in S \)) which satisfy the relation \( (x \oplus x) \oplus A_2 = A_0 \).

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