
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 company that manufactures energy-saving environmental protection equipment has a supply shortage for a particular device. The monthly production volume $x$ (in units) and the unit selling price $y_{1}$ (in ten thousand yuan) satisfy the relationship $y_{1}=150- \frac {3}{2}x$, with a minimum unit selling price of 90 ten thousand yuan. The monthly production volume $x$ (in units) and the total production cost $y_{2}$ (in ten thousand yuan) satisfy the relationship $y_{2}=600+72x$. How many units should be produced monthly to maximize the average profit per unit? What is the maximum average profit?

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