
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, B, and C form a team to participate in a team game challenge. Each of them attempts the challenge independently. The probability of A successfully completing the challenge is $\frac{1}{3}$, the probability of both A and B successfully completing the challenge is $\frac{1}{6}$, and the probability of both B and C successfully completing the challenge is $\frac{1}{5}$. Each successful challenge is awarded 2 points, and the sum of the points is considered the team's total score.
(I) Find the probability of B and C successfully completing the challenge independently;
(II) Find the probability that the team's total score is 4 points;
(III) If the team's total score is not less than 4 points, the team can participate in the next round. Find the probability that the team participates in the next round.

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