
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
Among twelve balls, 6 are white, 6 are black, and they are numbered from \(1, 2, 3, \ldots, 11, 12\).

(a) How many arrangements are possible where the balls are placed in a sequence such that always one white is followed by one black?

(b) How many arrangements are possible such that only whites occupy the first 6 places?

(c) How many different groups of seven can be formed such that there are always 4 whites and 3 blacks in the group?

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