
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
Suzhou is known as the "Cloud Capital of China," with the largest cloud computing data center in East China, a $CG$ animation rendering cluster, and becoming the fifth city in China to have a quantum communication node after Beijing, Shanghai, Hefei, and Jinan. In order to calculate the computing power of the Intelligent Computing Center, a certain number of large data centers ($n$) and $5$ small data centers are randomly selected for computing power analysis. The probability of selecting $2$ small data centers at once is $\frac{2}{9}$. <br/> $(1)$ Find the value of $n$; <br/> $(2)$ If $3$ data centers are selected at once, assuming the number of small data centers selected is $X$, find the distribution and mathematical expectation of $X$.

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