
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 20-letter arrangements of 6 A's, 7 B's, and 7 C's have no A's in the first 7 letters and no B's in the last 7 letters?
A) $2^7 \cdot 3^6$
B) $\sum_{x=0}^{\min(7,7-y)} \binom{7}{x} \binom{7}{y} \binom{6}{6-y, 7-x, 7-x-y}$
C) $\sum_{x=0}^{6}\binom{6}{x}^2 \binom{8}{8-x}$
D) $\frac{20!}{6!7!7!}$

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