
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 of the following expressions:
(1) $\sum_{k=0}^{n} \frac{3^{k+1}}{k+1}\binom{n}{k}$.
(2) $\sum_{k=2}^{n} \binom{k}{k-2} \quad(n \geq 2)$.
(3) $\sum_{k=0}^{n} (k^{2}+3)\binom{n}{k}$.
(4) $\sum_{k=0}^{n} (k+1) 2^{k}\binom{n}{k}$.
(5) $\sum_{k=0}^{n} \frac{(-1)^{k}}{k+1}\binom{n}{k} \quad(n \geq 1)$.
(6) $\sum_{k=0}^{n} \frac{k+3}{k+1}\binom{n}{k}$.

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