
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 certain high school held a graduation ceremony for senior students on May 20th. To create a relaxed atmosphere for the seniors, there was a "blind box" game segment during the ceremony. The host presented 10 blind boxes, each containing a model of a school landmark building. Among them, there were 3 "campus" models, 4 "library" models, 2 "hall of fame" models, and 1 "science museum" model.  
$(1)$ Draw 2 blind boxes at once. Find the probability that the 2 blind boxes contain the same type of model.  
$(2)$ Draw 2 blind boxes successively without replacement. Find the probability that the second box drawn is a "library" model.  
$(3)$ Student A is a "science enthusiast" and really wants to get the "science museum" model. To fulfill A's wish, the host designed the following game rules: There are 10 identical balls in an opaque bag, including 9 white balls and 1 red ball. In each draw with replacement, if the red ball is drawn, A can take the "science museum" model and the game ends. Now let A participate in the game, and A is allowed to draw a maximum of 10 times following the game rules. If A obtains the "science museum" model after the X-th draw $(X=1,2,\ldots,10)$, find the distribution of X.

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