
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 have no other differences except for color). Person A draws 1 ball from the bag. If a white ball is drawn, the drawn white ball is returned to the bag, and then 1 yellow ball is taken out and replaced with 1 white ball; if a yellow ball is drawn, the drawn yellow ball is returned to the bag, and then 1 white ball is taken out and replaced with 1 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 draws balls 4 times, and the number of white balls in the bag is denoted as $X$.  
$(1)$ Find the probability that there is only one color of balls in the bag.  
$(2)$ Find the distribution and expectation of $X$.

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