
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
For the following functions \( f(n) \), find \( \Delta^{k} f(n) \).

1. \( f(n) = 2^{n} + (-1)^{n} \).
2. \( f(n) = \frac{1}{n} \).
3. \( f(n) = n \cdot 2^{n} \).
4. \( f(n) = \cos (n x + a) \).

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