
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 $M$ be a positive multiple of 6. One blue ball and $M$ yellow balls are arranged in a line at random. Define $Q(M)$ as the probability that at least $\tfrac{2}{3}$ of the yellow balls are on the same side of the blue ball. Calculate the first value of $M$ such that $Q(M) < \tfrac{320}{450}$ and determine the sum of its digits.
A) 11
B) 12
C) 13
D) 14
E) 15

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