
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 company sells a newly developed "Smart Star" robot for tutoring students in two locations, A and B, simultaneously. The production cost of this robot is $200 per unit. The selling price, sales volume, and various expenses such as advertising and management costs in locations A and B are shown in the table below:

|           | Monthly Sales Volume $x$ (units) | Selling Price $a$ (dollars per unit) | Monthly Advertising, Management, etc. Costs (dollars per month) |
|-----------|-----------------------------------|--------------------------------------|------------------------------------------------------------------|
| Location A | $x$                               | $a=500$                              | $100x+10000$                                                     |
| Location B | $x$                               | $a=1200-x$                           | $50000$                                                          |

$(1)$ If the monthly sales profits in locations A and B are $w_{1}$ and $w_{2}$ dollars, respectively, find the functional relationships between $w_{1}$ and $x$ and between $w_{2}$ and $x$.

$(2)$ If a total of 1000 units are sold in locations A and B each month, how should the sales volumes in locations A and B be arranged to maximize profit?

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