
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
From July 6th to 14th, 2022, the 29th International Congress of Mathematicians, known as the "Mathematics Olympics" in the mathematical community, was held online due to the pandemic. Mathematicians from all over the world gathered in the cloud to participate in the event. A math enthusiast association at a school randomly surveyed 100 students and obtained the following results: 55% of the surveyed students were male, 40 students liked mathematics, and the rest did not like mathematics; among the surveyed female students, 20 liked mathematics, and the rest did not like mathematics. 

$(1)$ Please complete the following $2\times 2$ contingency table:

|          | Likes Mathematics | Does Not Like Mathematics | Total |
|----------|-------------------|---------------------------|-------|
| Male     |                   |                           |       |
| Female   |                   |                           |       |
| Total    |                   |                           |       |

$(2)$ Based on the $2\times 2$ contingency table, determine whether there is a 99.5% confidence that the preference for mathematics among students at this school is related to their gender.

Reference formula: ${K}^{2}=\frac{{n{{(ad-bc)}^2}}}{{(a+b)(c+d)(a+c)(b+d)}}$, where $n=a+b+c+d$.

Critical value table:

| $P(K^{2}\geqslant k_{0})$ | 0.10 | 0.05 | 0.01 | 0.005 | 0.001 |
|---------------------------|------|------|------|-------|-------|
| $k_{0}$                   | 2.706| 3.841| 6.635| 7.879 | 10.828|

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