
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 factory produces two types of products, $I$ and $II$, and packages them in three different styles: $A$, $B$, and $C$. The weights of the packages and the weights of $I$ and $II$ products contained in each package are shown in the table below:

| Package Style | Weight (tons) | Weight of $I$ product (tons) | Weight of $II$ product (tons) |
|---------------|---------------|------------------------------|-------------------------------|
| $A$           | $6$           | $3$                          | $3$                           |
| $B$           | $5$           | $3$                          | $2$                           |
| $C$           | $5$           | $2$                          | $3$                           |

Now, a truck with a maximum load capacity of $28$ tons is used to transport $5$ packages at a time, with at least $1$ of each style.

$(1)$ If exactly $28$ tons of packaged products are transported, then the numbers of $A$, $B$, and $C$ packages in the transport plan are ____.

$(2)$ If the weight of $I$ products transported does not exceed $13$ tons, and the weight of $II$ products transported is maximized, then the numbers of $A$, $B$, and $C$ packages in the transport plan are ____.

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