
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
A community organized a Beijing Olympics knowledge promotion activity. The "drawing cards with prizes" game particularly attracted attention. The game rules are as follows: There are 8 cards of the same size and shape in a box, with either an "Olympic mascot" or an "Olympic emblem" printed on them. Four people participate in the game and take turns drawing cards from the box, two cards at a time, without putting them back. The game ends and the person wins a prize when one of the four people draws two "Olympic mascot" cards at once.

(1) Before the game started, a high school student asked: How many "Olympic emblem" cards are there in the box? The host said: If two cards are randomly drawn from the box, the probability that both are not "Olympic emblem" cards is $\frac{25}{28}$. How many "Olympic emblem" cards are there in the box?

(2) Suppose four people, A, B, C, and D, participate in the game and draw cards in turn. Let $\xi$ represent the total number of times cards are drawn when someone wins a prize and ends the game. Find the probability distribution and mathematical expectation of $\xi$.

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