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In a certain tourist area, the number of people engaging in tourism service work each month of the year approximately follows a periodic pattern. Thus, the number of people, $f(n)$, engaging in tourism service work in the $n$-th month can be approximately described by the function:

\[ f(n) = 100[A \cos (\omega n + \alpha) + k] \]

where $n$ is a positive integer representing the month, $A$ and $k$ are positive integers, $\omega > 0$, and $\alpha \in (\frac{\pi}{2}, \pi)$.

Statistics reveal the following patterns for the number of people engaging in tourism service work each month:
(i) The number of people in the same month each year is roughly the same.
(ii) The difference in the number of people between the peak in August and the trough in February is about 400 people.
(iii) The number of people engaging in tourism service work in February is approximately 100 and increases each month, reaching a peak in August.

(1) Based on the given information, try to determine an expression for $f(n)$ that meets the conditions.
(2) Generally, when the number of people engaging in tourism service work in the area reaches or exceeds 400, the area enters the "peak tourism season" for the year. Determine which months of the year are considered peak tourism season in the area, and explain the reasoning.

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