
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 development of new energy can effectively solve the energy shortage and environmental pollution caused by the use of traditional energy sources in China. The introduction of national new energy policies has brought spring to the new energy industry. It is known that a new energy company in Zhejiang has an annual fixed cost of $6$ million. For each production of $x$ (where $x\in N*$) equipment, an additional cost of $t$ million is required. If the annual output is less than $100$ units, then $t=\frac{1}{2}{x^2}+60x$; if the annual output is not less than $100$ units, then $t=152x+\frac{{24200}}{x}-4700$. The selling price of each equipment is $1.5$ million. Through market analysis, all equipment produced by the company can be sold.

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

$(2)$ For how many units of annual output will the company achieve maximum profit?

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