
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 school organizes a badminton competition, with each match adopting a best-of-five format (the match does not end in a tie, and the first player to win three games wins the match). The probability of each player winning the first game is $\frac{1}{2}$. Starting from the second game, the probability of winning each game is influenced by the result of the previous game. If a player won the previous game, the probability of winning the current game is $\frac{1+p}{2}$; if a player did not win the previous game, the probability of winning the current game is $\frac{1-p}{2}$. The probability of one player winning the first two consecutive games is $\frac{5}{16}$. Find the value of $p$ and the probability of the match ending after $4$ games.

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