
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 order to promote the comprehensive development of students' morality, intelligence, physical fitness, aesthetics, and labor skills, a school has established a biotechnology group to plant three types of crops $A$, $B$, and $C$ alternately in the same experimental field (only one type of crop can be planted in the field each time). To maintain soil fertility, each type of crop cannot be planted continuously, and a total of three plantings are done. After planting crop $A$ each time, there is a $\frac{1}{3}$ probability of planting $B$ and a $\frac{2}{3}$ probability of planting $C$; under the condition of planting $B$ each time, the probability of planting $A$ again is $\frac{1}{4}$, and the probability of planting $C$ is $\frac{3}{4}$; under the condition of planting $C$ each time, the probability of planting $A$ again is $\frac{2}{5}$, and the probability of planting $B$ is $\frac{3}{5}$.

$(1)$ Given that crop $B$ is planted for the first time, find the probability of planting crop $A$ for the third time.

$(2)$ Given that crop $A$ is planted for the first time, find the distribution and expectation of the number of times crop $A$ is planted, denoted as $X$.

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