
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
In recent years, handicrafts made by a certain handicraft village have been very popular overseas. The villagers of the village have established a cooperative for exporting handicrafts. In order to strictly control the quality, the cooperative invites 3 experts to inspect each handicraft made by the villagers. The quality control process is as follows: $(i)$ If all 3 experts consider a handicraft to be of good quality, then the quality of the handicraft is rated as grade $A$; $(ii)$ If only 1 expert considers the quality to be unsatisfactory, then the other 2 experts will conduct a second quality check. If both experts in the second check consider the quality to be good, then the handicraft is rated as grade $B$. If one or both of the experts in the second check consider the quality to be unsatisfactory, then the handicraft is rated as grade $C$; $(iii)$ If 2 or all 3 experts consider the quality to be unsatisfactory, then the handicraft is rated as grade $D$. It is known that the probability of 1 handicraft being considered unsatisfactory by 1 expert in each quality check is $\frac{1}{3}$, and the quality of each handicraft is independent of each other. Find: $(1)$ the probability that 1 handicraft is rated as grade $B$; $(2)$ if handicrafts rated as grade $A$, $B$, and $C$ can be exported with profits of $900$ yuan, $600$ yuan, and $300$ yuan respectively, while grade $D$ cannot be exported and has a profit of $100$ yuan. Find: $①$ the most likely number of handicrafts out of 10 that cannot be exported; $②$ if the profit of 1 handicraft is $X$ yuan, find the distribution and mean of $X$.

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