
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
(1) A factory produces three different models of products, $A$, $B$, and $C$, with the ratio of their quantities being $2:3:5$. Now, using stratified sampling, a sample of size $n$ is drawn, and the sample contains $16$ units of model $A$. Then, the sample size $n = \_\_\_\_\_\_\_\_$.

(2) If a real number $a$ is randomly selected from the interval $[-5,5]$, then the probability that the line $x+y+a=0$ intersects with the circle $(x-1)^2+(y+2)^2=2$ is $\_\_\_\_\_\_\_\_$.

(3) As shown in the algorithm, the number of times the loop is executed is $\_\_\_\_\_\_\_\_$.

```
$\boxed{\begin{matrix}
s=0 \\
i=1 \\
DO \\
s=s+i \\
i=i+1 \\
LOOP\quad UNTIL\quad i > 1 000
\end{matrix}}$
```

(4) In two boxes, labeled $A$ and $B$, there are four balls each, labeled $1$, $2$, $3$, and $4$. If one ball is drawn from each box, with each ball having an equal chance of being drawn, then the probability that the sum of the numbers on the two balls is divisible by $3$ is $\_\_\_\_\_\_\_\_$.

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