
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
The $5G$ technology is playing an increasingly important role in the national economy. A certain technology company decides to develop and produce a new $5G$ machine to meet the needs of a certain $5G$ application. The monthly fixed cost of producing this machine is $4$ million yuan. In addition to producing $x$ units, an additional cost of $p(x)$ million yuan is required. When the monthly output is less than $70$ units, $p(x)=\frac{1}{2}x^2+40x$ (million yuan); when the monthly output is not less than $70$ units, $p(x)=101x+\frac{6400}{x}-2060$ (million yuan). If each machine is sold for $100$ million yuan, and all the machines can be sold.<br/>$(1)$ Find the function relationship of monthly profit $y$ (million yuan) with monthly output $x$ (units);<br/>$(2)$ How many units should be produced per month for the company to achieve the maximum monthly profit? And find the maximum monthly profit.

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