
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
How many $12$-letter arrangements of $4$ A's, $4$ B's, and $4$ C's have no A's in the first $4$ letters, no B's in the next $4$ letters, and no C's in the last $4$ letters?
A) $\sum_{k=0}^4 \binom{4}{k}$  
B) $\sum_{k=0}^4 \binom{4}{k}^2$  
C) $\sum_{k=0}^4 \binom{4}{k}^3$  
D) $4^6$  
E) $\sum_{k=0}^4 2^{k}$

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