
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
Supermarkets $A$ and $B$ each have 15 tons of mangoes. They are now transporting mangoes from $A$ and $B$ to two destinations, $X$ and $Y$, respectively. Destination $X$ requires 16 tons of mangoes, and destination $Y$ requires 14 tons of mangoes. The transportation cost from $A$ to destination $X$ is $50$ dollars per ton, and to destination $Y$ is $30$ dollars per ton. The transportation cost from $B$ to destination $X$ is $60$ dollars per ton, and to destination $Y$ is $45$ dollars per ton. 

$(1)$ Let $x$ tons of mangoes be transported from $A$ to destination $X$. Please complete the table below:

|          | Destination $X$ (tons) | Destination $Y$ (tons) |
|----------|------------------------|------------------------|
| $A$      | $x$                    | ______                 |
| $B$      | ______                 | ______                 |

$(2)$ Let the total transportation cost be $y$ dollars. Write down the relationship between $y$ and $x$, and directly state the range of values for $x$.

$(3)$ How should the mangoes be transported to minimize the transportation cost?

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