
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]


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The fat content (unit: %) refers to the proportion of fat weight to the total body weight. A sports physiologist conducted a survey on the fat content of participants in a fitness activity using stratified random sampling with sample size ratio. If the sample data is unknown, but it is known that $120$ males were sampled with an average and variance of $14$ and $6$ respectively, and $90$ females were sampled with an average and variance of $21$ and $17$ respectively.  
$(1)$ Calculate the mean and variance of the total sample based on this data, and estimate the mean and variance of the fat content of all participants in the fitness activity. (Results should be rounded to the nearest integer)  
$(2)$ Assuming that the fat content of all participants is a random variable $X$, and $X\sim N\left(\mu ,23\right)$, where $\mu$ is approximately the total sample mean calculated in $(1)$. Now, randomly select $4$ participants from all participants, and find the probability that the fat content of the $4$ participants is all less than $12.2\%$.  
Given: If a random variable $X$ follows a normal distribution $N\left(\mu ,\sigma \right)$, then $P\left(\mu -\sigma \right)\leqslant X\leqslant \mu +\sigma \approx 0.6827$, $P\left(\mu -2\sigma \right)\leqslant X\leqslant \mu +2\sigma \approx 0.9545$, $\sqrt{22}\approx 4.7$, $\sqrt{23}\approx 4.8$, $0.15865^{3}\approx 0.004$, $0.15865^{4}\approx 0.0006$.

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