
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
City $A$ and city $B$ have inventories of a certain type of machine with 12 units and 8 units, respectively. It is decided to support city $C$ with 10 units and city $D$ with 10 units. The transportation cost for moving a machine from city $A$ to city $C$ or city $D$ is $300$ yuan and $500$ yuan, respectively. The transportation cost for moving a machine from city $B$ to city $C$ or city $D$ is $400$ yuan and $800$ yuan, respectively.

$(1)$ Let $x$ represent the number of machines transported from city $A$ to city $C$. Find the function relationship of the total transportation cost $y$ in terms of $x$.

$(2)$ Determine the range of values for the independent variable $x$.

$(3)$ Design a transportation plan with the lowest total transportation cost and calculate this minimum cost.

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