
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 fencing, you win a round if you are the first to reach  $15$  points. Suppose that when  $A$  plays against  $B$ , at any point during the round,  $A$  scores the next point with probability  $p$  and  $B$  scores the next point with probability  $q=1-p$ . (However, they never can both score a point at the same time.)

Suppose that in this round,  $A$  already has  $14-k$  points, and  $B$  has  $14-\ell$  (where  $0\le k,\ell\le 14$ ). By how much will the probability that  $A$  wins the round increase if  $A$  scores the next point?

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