
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 a certain high school, there are 1000 freshmen, including 600 boys and 400 girls. In order to obtain the height information of all freshmen in the school, a stratified random sampling with a sample size ratio allocation is used to select a sample of size 50.  
$(1)$ Find the number of boys and girls selected in the sample.  
$(2)$ After observing the indicator values (unit: cm) of the sample, it is found that the mean height of boys in the sample is 170 with a variance of 14, and the mean height of girls in the sample is 160 with a variance of 34. Calculate the total variance of the sample and estimate the variance of the height of all freshmen in the first year.

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