
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 a fencing match, the first to reach 15 hits wins. Suppose that in a match between competitors $A$ and $B$, the probability that $A$ scores the next hit is $p$ and the probability that $B$ scores the next hit is $q = 1 - p$. They never hit simultaneously.

Suppose that in this match, $A$ has already scored $14 - k$ hits and $B$ has scored $14 - \ell$ hits, where $k$ and $\ell$ are non-negative integers less than 15. If $A$ scores one more hit, by how much does the probability of $A$ ultimately winning the match increase?

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