
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 class, there are $6$ boys and $4$ girls. $5$ students are randomly selected to form a math interest group, and another $5$ students are selected to form a physics interest group. <br/>$(1)$ Find the probability that the math interest group includes boy $A$ but not girl $a$; <br/>$(2)$ Let $X$ represent the number of girls in the physics interest group. Find the distribution table of $X$ and the mathematical expectation $E(X)$.

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