
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

...

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According to the franchise operation agreement between the Beijing Organizing Committee for the 2022 Winter Olympics and the franchise manufacturer, various franchise goods for the 2022 Beijing Winter Olympics, including Bing Dwen Dwen dolls, will cease production starting from July 1st. The sales data statistics table of two colors of Bing Dwen Dwen sold by a certain retail store on a certain day (before July 1st) morning are given below (assuming each person can only buy one Bing Dwen Dwen):

|         | Blue     | Pink     |
|---------|----------|----------|
| Male    | $\frac{5a}{6}$ | $\frac{a}{6}$ |
| Female  | $\frac{2a}{3}$ | $\frac{4a}{3}$ |

$(1)$ If there is a $99\%$ certainty that the color of Bing Dwen Dwen purchased by customers is related to their gender, find the minimum value of $a$;  
$(2)$ Under the condition that $a$ achieves the minimum value, now select $p$ people from customers who buy blue Bing Dwen Dwen and $q$ people from customers who buy pink Bing Dwen Dwen, with $p+q=9 (p,q\geqslant 0)$. Let $X$ denote the number of female customers selected. Find the distribution table and mathematical expectation of $X$.  
Given: ${K^2}=\frac{{n{{(ad-bc)}^2}}}{{(a+b)(c+d)(a+c)(b+d)}}$  

| $P(K^{2}\geqslant k)$ | $0.05$ | $0.010$ | $0.001$ |
|-----------------------|--------|---------|---------|
| $k$                   | $3.841$| $6.635$ | $10.828$|

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