
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
Find the sums:

a) \(1 \cdot n + 2(n-1) + 3(n-2) + \ldots + n \cdot 1\).

b) \(S_{n, k} = (1 \cdot 2 \cdot \ldots \cdot k) \cdot (n(n-1) \ldots (n-k+1)) + (2 \cdot 3 \cdot \ldots \cdot (k+1)) \cdot ((n-1)(n-2) \ldots (n-k)) + \ldots + ((n-k+1)(n-k+2) \ldots n) \cdot (k(k-1) \ldots 1)\).

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