
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
Players A and B participate in a two-project competition, with each project adopting a best-of-five format (the first player to win 3 games wins the match, and the competition ends), and there are no ties in each game. Based on the statistics of their previous matches, player A has a probability of $\frac{2}{3}$ of winning each game in project $A$, and a probability of $\frac{1}{2}$ of winning each game in project $B$, with no influence between games.  
$(1)$ Find the probability of player A winning in project $A$ and project $B$ respectively.  
$(2)$ Let $X$ be the number of projects player A wins. Find the distribution and mathematical expectation of $X$.

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