
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
Let $N$ be a positive multiple of $6$. One blue ball and $N$ yellow balls are arranged linearly in a random order. Define $P(N)$ as the probability that at least $\tfrac{2}{3}$ of the yellow balls are on the same side of the blue ball. Given that $P(6)=1$ and $P(N)$ approaches $\tfrac{5}{6}$ as $N$ becomes large, calculate the sum of the digits of the largest value of $N$ such that $P(N) > \tfrac{3}{4}$.
A) 4
B) 6
C) 8
D) 10

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