
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 $S=\{A_0, A_1, A_2\}$, and define the operation $\oplus$ on $S$ as: $A_i \oplus A_j = A_k$, where $k$ is the remainder of $i+j$ divided by 3, with $i, j \in \{1, 2, 3\}$. Determine the total number of ordered pairs $(i, j)$ that satisfy the equation $(A_i \oplus A_j) \oplus A_i = A_0$.

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