
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
In a certain tourist area, the number of visitors received each month of the year approximately follows a periodic pattern. Therefore, the number of people working in tourism services in the $n$-th month, $f(n)$, can be approximately described by the function $f(n)=A\cos (wn+\theta)+k$, where the positive integer $n$ represents the month and $n\in[1,12]$, for example, $n=1$ represents January. $A$ and $k$ are positive integers, $w > 0$, and $\theta\in(0,\pi)$. It has been found that the number of people working in tourism services in this area each month of the year follows the following patterns:
1. The number of people working in tourism services in the same month each year is basically the same;
2. The difference between the number of people working in tourism services in August, the month with the most workers, and February, the month with the least, is 400 people;
3. In February, the number of people working in tourism services in the area is 100, and it increases month by month until it reaches the maximum in August.
   1. Based on the information provided, find the expression for $f(n)$;
   2. Generally, when the number of people working in tourism services in the area is 400 or more, the area also enters the "peak season" of tourism for the year. Which months of the year are the "peak season" for this area? Please explain your reasoning.

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