
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 \( S = \left\{ A_{0}, A_{1}, A_{2}, A_{3} \right\} \). Define the operation "\(\oplus\)" on \(S\) as follows: \( A_{i} \oplus A_{j} = A_{k} \), where \( k \) is the remainder when \( i + j \) is divided by 4, for \( i, j = 0, 1, 2, 3 \). Determine the number of elements \( x (x \in S) \) that satisfy the relation \( (x \oplus x) \oplus A_{2} = A_{0} \).

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