
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 certain factory is currently producing $50$ more machines per day on average than originally planned. The time required to produce $600$ machines now is the same as the time required to produce $450$ machines according to the original plan.  
$(1)$ Find the average number of machines produced per day now.  
$(2)$ The current cost of producing each machine is $600$ yuan, and the selling price is $900$ yuan. All machines produced per day are sold out. The factory decides to adjust the price. Market research indicates that increasing the price of each machine by $50$ yuan results in selling $20$ fewer machines per day. What price should the factory set to maximize daily profit?

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