
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 $S=\{A, A_1, A_2, A_3, A_4, A_5\}$. Define the operation "$\oplus$" on $S$ as: $A_i \oplus A_j = A_k$, where $k$ is the remainder of $i+j$ divided by 4, with $i, j = 0, 1, 2, 3, 4, 5$. Then, the number of $x$ (where $x \in S$) that satisfies the equation $(x \oplus x) \oplus A_2 = A$ is.

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