
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
(a) A bag contains 11 balls numbered from 1 to 11, 7 of which are black and 4 of which are gold. Julio removes two balls as described above. What is the probability that both balls are black?

(b) For some integer \( g \geq 2 \), a second bag contains 6 black balls and \( g \) gold balls; the balls are numbered from 1 to \( g+6 \). Julio removes two balls as described above. The probability that both balls are black is \( \frac{1}{8} \). Determine the value of \( g \).

(c) For some integer \( x \geq 2 \), a third bag contains \( 2x \) black balls and \( x \) gold balls; the balls are numbered from 1 to \( 3x \). Julio removes two balls as described above. The probability that both balls are black is \( \frac{7}{16} \). Determine the value of \( x \).

(d) For some integer \( r \geq 3 \), a fourth bag contains 10 black balls, 18 gold balls, and \( r \) red balls; the balls are numbered from 1 to \( r+28 \). This time, Julio draws three balls one after another. The probability that two of these three balls are black and one of these three balls is gold is at least \( \frac{1}{3000} \). What is the largest possible value of \( r \)?

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