
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
Consider a sequence of balls, including one red ball, $k$ blue balls, and $N$ green balls, where $N + k$ is divisible by $4$. Arrange these balls in a line in random order. Let $Q(N, k)$ denote the probability that no blue ball is adjacent to the red ball. If $Q(4, 16) = 1$ and $Q(N, k)$ approaches $\frac{7}{8}$ as $N + k$ increases, determine the smallest sum of $k$ and $N$ for which $Q(N, k) < \frac{7}{9}$.
A) 3
B) 4
C) 5
D) 6
E) 7

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