
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
There are $20$ boxes numbered $1$, $2$, $3$, $\ldots$, $18$, $19$, $20$. The first box contains $2$ yellow balls and $1$ green ball, while the rest of the boxes contain $2$ yellow balls and $2$ green balls. Now, a ball is taken from the first box and placed into the second box, then a ball is taken from the second box and placed into the third box, and so on until a ball is taken from the $19$th box and placed into the $20$th box. Let $p_{i}$ denote the probability of drawing a yellow ball from the $i$th box.
$(1)$ Find $p_{2}$ and $p_{3}$.
$(2)$ Find $p_{20}$.

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