
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 university needs $40L$ of helium gas to make balloon decorations for its centennial celebration. The chemistry club voluntarily took on this task. The club's equipment can produce a maximum of $8L$ of helium gas per day. According to the plan, the club must complete the production within 30 days. Upon receiving the task, the club members immediately started producing helium gas at a rate of $xL$ per day. It is known that the cost of raw materials for producing $1L$ of helium gas is $100$ yuan. If the daily production of helium gas is less than $4L$, the additional cost per day is $W_1=4x^2+16$ yuan. If the daily production of helium gas is greater than or equal to $4L$, the additional cost per day is $W_2=17x+\frac{9}{x}-3$ yuan. The production cost consists of raw material cost and additional cost.

$(1)$ Write the relationship between the total cost $W$ (in yuan) and the daily production $x$ (in $L$).

$(2)$ When the club produces how many liters of helium gas per day, the total cost is minimized? What is the minimum cost?

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