
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 produces large air purification equipment with fixed annual costs of $5$ million dollars. For each production of $x$ ($x\in N^{*}$) units of equipment, an additional cost of $t$ million dollars is required. If the annual production is less than $150$ units, then $t=\frac{1}{2}{x}^{2}+128x$; if the annual production is not less than $150$ units, then $t=210x+\frac{400000}{x}-6900$. The selling price of each equipment is $2$ million dollars. Through market analysis, it is determined that all the equipment produced by the company can be sold.

$(1)$ Write the relationship between the annual profit $y$ (in million dollars) and the annual production $x$ (in units).

$(2)$ At what annual production level does the company achieve maximum profit?

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