
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 product has three quality indicators, denoted as $x$, $y$, and $z$. The product's grade is evaluated using the composite indicator $S=x+y+z$. If $S\leqslant 4$, the product is classified as first-class. Now, from a batch of this product, $10$ items are randomly selected as a sample, and their quality indicators are listed below:

| Product Number | $A_{1}$ | $A_{2}$ | $A_{3}$ | $A_{4}$ | $A_{5}$ |
|----------------|---------|---------|---------|---------|---------|
| Quality Indicators $(x,y,z)$ | $(1,1,2)$ | $(2,1,1)$ | $(2,2,2)$ | $(1,1,1)$ | $(1,2,1)$ |

| Product Number | $A_{6}$ | $A_{7}$ | $A_{8}$ | $A_{9}$ | $A_{10}$ |
|----------------|---------|---------|---------|---------|----------|
| Quality Indicators $(x,y,z)$ | $(1,2,2)$ | $(2,1,1)$ | $(2,2,1)$ | $(1,1,1)$ | $(2,1,2)$ |

1. Utilize the sample data provided in the table to estimate the first-class rate of this batch of products.
2. From the first-class items in the sample, randomly select $2$ items:
   - List all possible outcomes based on the product numbers.
   - Define event $B$ as "both selected items have a composite indicator $S$ equal to $4$." Calculate the probability of event $B$ occurring.

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