
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 red ball and $M$ green balls are arranged in a line in random order. Let $Q(M)$ be the probability that at least $\tfrac{2}{3}$ of the green balls are on the same side of the red ball. Determine the smallest value of $M$ such that $Q(M) < \tfrac{3}{4}$.

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