
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
A travel agency has devised a plan for organizing group tours to Guangzhou by charter flights. The charter flight cost is $30,000, with a maximum of $75$ people in the tour group. The relationship between the number of people $x$ (in units) in the tour group and the fee $P(x)$ (in units of yuan) paid to the travel agency is as follows:
$$P(x) = \begin{cases} 1800, & 1 \leqslant x \leqslant 30, \text{ and } x \in \mathbb{N^*}, \\ -20x + 2400, & 30 < x \leqslant 75, \text{ and } x \in \mathbb{N^*}. \end{cases}$$

(1) Represent the profit $y$ (in units of yuan) of the travel agency as a function of the number of people $x$ in the tour group (Note: profit = fees collected - charter flight cost);

(2) How many people in the tour group would maximize the travel agency's profit? Calculate the maximum profit.

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