
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
Eggplant and cucumber are common vegetables in daily life with a long history of cultivation. A farmer has opened up 4 equally sized and uniform soil plots in front of his small courtyard. The 4 plots of land form a rectangular grid (2 rows and 2 columns) with the same soil quality. Each plot of land only plants either eggplant or cucumber, and all 4 plots are fully planted. If the probability of planting eggplant in each plot is $\frac{1}{3}$, with an emergence rate of 95% for eggplant seeds; and the probability of planting cucumber in each plot is $\frac{2}{3}$, with an emergence rate of 98% for cucumber seeds. The planting of each type of vegetable in each plot does not affect each other, and the germination of eggplant and cucumber seeds does not affect each other.  
$(1)$ $\left(i\right)$ Find the probability of exactly 2 plots planting eggplants;  
$(ii)$ Find the germination rate of seeds for each plot;  
$(2)$ If it is known that in each column there is a plot planting eggplants, let the number of rows planting cucumbers (as long as there is a cucumber planted in that row) be a random variable $X$, find the distribution table and the expected value of $X$.

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