
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
Given the set $S=\{A_0,A_1,A_2,A_3\}$, an operation $⊕$ is defined on $S$: $A_i⊕A_j=A_k$, where $k$ is the remainder of $i+j$ divided by $4$, $i$, $j=0$, $1$, $2$, $3$. Determine the number of ordered pairs $(i,j)$ that satisfy the relation $((A_i⊕A_i)⊕A_j)=A_0$.

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