
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
To investigate the growth inhibitory effect of a certain drug on mice, $40$ mice were divided into two groups, a control group (without the drug) and an experimental group (with the drug).<br/>$(1)$ Suppose the number of mice in the control group among two mice is $X$, find the probability distribution and mathematical expectation of $X$;<br/>$(2)$ The weights of the $40$ mice are as follows (unit: $g) (already arranged in ascending order):<br/>Control group: $17.3, 18.4, 20.1, 20.4, 21.5, 23.2, 24.6, 24.8, 25.0, 25.4$<br/>$26.1, 26.3, 26.4, 26.5, 26.8, 27.0, 27.4, 27.5, 27.6, 28.3$<br/>Experimental group: $5.4, 6.6, 6.8, 6.9, 7.8, 8.2, 9.4, 10.0, 10.4, 11.2$<br/>$14.4, 17.3, 19.2, 20.2, 23.6, 23.8, 24.5, 25.1, 25.2, 26.0$<br/>$(i)$ Find the median $m$ of the weights of the $40$ mice and complete the following $2\times 2$ contingency table:<br/><table><tbody><tr><td align="center"></td><td align="center">$ \lt m$</td><td align="center">$\geq m$</td></tr><tr><td align="center">Control Group</td><td align="center">_____</td><td align="center">_____</td></tr><tr><td align="center">Experimental Group</td><td align="center">_____</td><td align="center">_____</td></tr></tbody></table>$($ii)$ Based on the $2\times 2$ contingency table, can we be $95\%$ confident that the drug inhibits the growth of mice.<br/>Reference data:<br/><table><tbody><tr><td align="center">$k_{0}$</td><td align="center">$0.10$</td><td align="center">$0.05$</td><td align="center">$0.010$</td></tr><tr><td align="center">$P(k^{2}\ge k_{0})$</td><td align="center">$2.706$</td><td align="center">$3.841$</td><td align="center">$6.635$</td></tr></tbody></table>

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