
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
Badminton game rules:<br/>①$21$-point system, the winner of each rally scores $1$ point and serves next; <br/>②When the score reaches $20:20$, the side leading by $2$ points wins the game; <br/>When the score is tied at $29:29$, the first to reach $30$ points wins the game. After a fierce battle, the score between players A and B is $27:28$. At a crucial moment, player A wins a point, making the score $28:28$. At the critical moment, based on past performance statistics, the probability of player A winning a rally when serving is $0.4$, and the probability of player A winning a rally when not serving is $0.5$. The outcomes of each rally are independent of each other. <br/>$(1)$ After the score is tied at $28:28$, find the probability of the game ending after two more rallies; <br/>$(2)$ After the score is tied at $28:28$, find the probability of player A winning the game.

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