
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
It is known that the relationship between the daily production cost $C$ (in yuan) and the production quantity $n$ (in pairs) of a certain shoe factory is $C = 4000 + 50n$.

(1) Calculate the cost of producing 1000 pairs of shoes in one day;  
(2) If the production cost one day is 48,000 yuan, how many pairs of shoes were produced that day?  
(3) Assuming the selling price of each pair of shoes is 90 yuan, and all produced shoes are sold out, write the function relationship of the profit $P$ for that day with respect to the production quantity $n$ of that day, and calculate the minimum number of pairs of shoes that must be produced each day to avoid a loss?

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