
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
Given that there are 6 white balls and 4 yellow balls in an opaque bag (these balls are identical except for color). Person A draws one ball from the bag. If a white ball is drawn, the drawn white ball is returned to the bag, and then one yellow ball is taken out and replaced with a white ball. If a yellow ball is drawn, the drawn yellow ball is returned to the bag, and then one white ball is taken out and replaced with a yellow ball. After thorough mixing, a second ball is drawn, and this process continues until all the balls in the bag are of the same color. It is known that person A has drawn balls 4 times.
- $(1)$ Find the probability that all the balls in the bag are of the same color after 4 draws.
- $(2)$ Find the probability that there are 4 white balls in the bag.

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