
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
A certain school used proportional distribution and stratified random sampling to collect the height data of the first-year students. Some of the statistical data are shown in the table below:

| Gender | Sample Size | Sample Mean | Sample Variance |
|--------|-------------|-------------|-----------------|
| Male   | 100         | 170         | 22              |
| Female | 100         | 160         | 38              |

Estimate the average height and variance of all first-year students in the school.

(Note: According to the exploration in Exercise 9.2 of the second volume of the compulsory high school mathematics textbook, the following conclusion can be drawn: Given that the population is divided into two layers, through stratified random sampling, the sample sizes, sample means, and sample variances of each layer are denoted as $m$, $\overline{x}$, ${s}_{1}^{2}$; $n$, $\overline{y}$, ${s}_{2}^{2}$. Let the total sample mean be $\overline{ω}$ and the sample variance be $s^{2}$, then ${s}^{2}=\frac{1}{m+n}\{m[{s}_{1}^{2}+{(\overline{x}-\overline{ω})}^{2}]+n[{s}_{2}^{2}+{(\overline{y}-\overline{ω})}^{2}]\}$)

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