
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
Vicky starts with a list \((a, b, c, d)\) of four digits. Each digit is \(0, 1, 2\), or 3. Vicky enters the list into a machine to produce a new list \((w, x, y, z)\). In the new list, \(w\) is the number of \(0\mathrm{s}\) in the original list, while \(x, y\), and \(z\) are the numbers of \(1\mathrm{s}, 2\mathrm{s}\), and \(3\mathrm{s}\), respectively, in the original list. For example, if Vicky enters \((1,3,0,1)\), the machine produces \((1, 2, 0, 1)\).

(a) What does the machine produce when Vicky enters \((2,3,3,0)\)?

(b) Vicky enters \((a, b, c, d)\) and the machine produces the identical list \((a, b, c, d)\). Determine all possible values of \(b + 2c + 3d\).

(c) Determine all possible lists \((a, b, c, d)\) with the property that when Vicky enters \((a, b, c, d)\), the machine produces the identical list \((a, b, c, d)\).

(d) Vicky buys a new machine into which she can enter a list of ten digits. Each digit is \(0, 1, 2, 3, 4, 5, 6, 7, 8\), or 9. The machine produces a new list whose entries are, in order, the numbers of \(0\mathrm{s}, 1\mathrm{s}, 2\mathrm{s}, 3\mathrm{s}, 4\mathrm{s}, 5\mathrm{s}, 6\mathrm{s}, 7\mathrm{s}, 8\mathrm{s}\), and \(9\mathrm{s}\) in the original list. Determine all possible lists, \(L\), of ten digits with the property that when Vicky enters \(L\), the machine produces the identical list \(L\).

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