
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 recent years, with the continuous progress of technology, technological achievements have been increasing year by year, especially in aspects related to physics - optics, electronics, mechanics, electrical, etc., are increasing faster. In order to protect intellectual property rights, it is necessary to transform technological achievements into technological patents. This requires a large number of patent agents to engage in patent writing work, and physics graduate students are more favored by patent agent companies. By training physics graduate students, they can write patents in fields such as chemistry, biology, and medicine, while graduate students in other subjects can only write patents in their own field. A large patent agency company conducted a survey on the employment intentions of physics graduate students in the field of patent agency through a random questionnaire survey. The data obtained is as follows:

|         | Like | Dislike |
|---------|------|---------|
| Female  | 105  | 75      |
| Male    | 60   | 90      |

$(1)$ According to the independence test with $\alpha = 0.001$, can it be concluded that the employment intention of physics graduate students in the field of patent agency is related to gender?  
$(2)$ The patent agency company stratifies the male graduate students among these 150 people according to their employment intentions in the field of patent agency. They use stratified random sampling to select 5 people, and then randomly select 3 people from these 5 people to investigate their intended annual salary after graduation in the form of a questionnaire. Among these 3 people, there are $X$ people who like to work as patent agents. Find the distribution table and mathematical expectation of $X$.  
Below is the critical value table and reference formula:

| $\alpha$ | 0.10 | 0.05 | 0.01 | 0.005 | 0.001 |
|----------|------|------|------|-------|-------|
| $x_{\alpha}$ | 2.706 | 3.841 | 6.635 | 7.879 | 10.828 |

$\chi^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$.

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