
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
For any positive integer \( x \), define \( \text{Accident}(x) \) to be the set of ordered pairs \( (s, t) \) with \( s \in \{0, 2, 4, 5, 7, 9, 11\} \) and \( t \in \{1, 3, 6, 8, 10\} \) such that \( x + s - t \) is divisible by 12. For any nonnegative integer \( i \), let \( a_i \) denote the number of \( x \in \{0, 1, \ldots, 11\} \) for which \( |\text{Accident}(x)| = i \). Find \( a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + a_{5}^{2} \).

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