
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
There are three cards in a box: one card has red on both sides, one card has black on both sides, and the remaining card has one red side and one black side.  
$(1)$ If a card is randomly drawn and one side is randomly shown, with the shown side being red, what is the probability that the other side is also red?  
$(2)$ If a card is randomly drawn, one side is shown, returned to the box, and then another card is drawn and one side is shown, the number of times black is shown is denoted as $X$. Find the distribution and expected value of the random variable $X$.

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