
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
Let \( \mathbb{R} \) be the reals. Let \( f : \mathbb{R} \to \mathbb{R} \) be an infinitely differentiable function such that \( f \left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1} \) for \( n = 1, 2, 3, \ldots \). Find the value of the derivatives of \( f \) at zero: \( f^{(k)}(0) \) for \( k = 1, 2, 3, \ldots \).

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