
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
In an $11$-point table tennis match, each winning point earns $1$ point. When a game reaches a tie at $10:10$, the serve alternates after each point, and the player who first gains a lead of $2$ points wins the game. Player A and player B are playing a singles match. It is assumed that when player A serves, the probability of player A scoring is $p$, and when player B serves, the probability of player A scoring is $\frac{2}{5}$. The outcomes of each point are independent. It is known that after a tie at $10:10$, player A serves first.  
$(1)$ If the probability of the game ending after two more points played by both players is $\frac{7}{15}$, find the value of $p$.  
$(2)$ Under the conditions in $(1)$, find the probability of player A winning after four more points played by both players.

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