
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 charters a plane for a tour group, where the charter fee for the travel agency is ¥15,000. The cost of each plane ticket for members of the tour group is settled with the travel agency as follows: if the number of people in the tour group is 30 or fewer, each plane ticket costs ¥900; if the number of people in the tour group exceeds 30, a discount is given, with the cost of each ticket decreasing by ¥10 for each additional person, but the maximum number of people in the tour group is 75. Let the number of people in the tour group be $x$, the price of each plane ticket be $y$ yuan, and the profit obtained by the travel agency be $W$ yuan.  
(1) Write the function relationship between $y$ and $x$;  
(2) Write the function relationship between $W$ and $x$;  
(3) When the number of people in the tour group is what, does the travel agency obtain the maximum profit? What is the maximum profit?

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