
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
The value of $\pi$ is an infinite non-repeating decimal. Historically, Chinese mathematicians such as Zu Chongzhi and Liu Hui, as well as foreign mathematicians like Viète and Euler, have conducted in-depth research on $\pi$. Currently, supercomputers have calculated the decimal part of $x$ to exceed 31.4 trillion digits. Scholars have found that as the number of decimal places of $\pi$ increases, the frequencies of the digits $0$ to $9$ tend to stabilize and become close to equal.  
$(1)$ If a digit is randomly selected from the decimal part of $\pi$, estimate the probability that the digit is even to be ______.  
$(2)$ A school is planning campus cultural development and intends to randomly select $2$ portraits from the above $4$ mathematicians. Using a list or tree diagram, calculate the probability that the selected portraits are exactly one Chinese and one foreign mathematician.

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