
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
Agricultural researchers randomly selected $50$ plants of a certain crop in an experimental field for research. The frequency distribution of the mass (in grams) of a single plant falling into each group is shown in the table below:

| Mass of a Single Plant (in grams) | Frequency |
|-----------------------------------|-----------|
| $[12.5,15.5)$                      | $2$       |
| $[15.5,18.5)$                      | $5$       |
| $[18.5,21.5)$                      | $11$      |
| $[21.5,24.5)$                      | $14$      |
| $[24.5,27.5)$                      | $11$      |
| $[27.5,30.5)$                      | $4$       |
| $[30.5,33.5]$                      | $3$       |

$(1)$ Based on the frequency distribution table, calculate the probability that the mass of a single plant of this crop falls in the interval $[27.5,33.5]$ (using frequency to estimate probability).  
$(2)$ Calculate the sample mean of the mass of these $50$ plants, denoted by $\overline{x}$. (Represent each group of data by the midpoint of the interval)  
$(3)$ If the mass $X$ of a single plant of this crop follows a normal distribution $N(\mu, \sigma^2)$, where $\mu$ is approximately the sample mean $\overline{x}$ and $\sigma^2$ is approximately the sample variance $S^2$, and it is calculated that $S^2 = 18.5364$, find $P(X < 27.37)$.  
Note:  
1. If $X$ follows a normal distribution $N(\mu, \sigma^2)$, then $P(\mu - \sigma < X < \mu + \sigma) = 0.6826$, $P(\mu - 2\sigma < X < \mu + 2\sigma) = 0.9544$;  
2. $\sqrt{18.5364} \approx 4.31$.

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