
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
The  $ 52$  cards in a deck are numbered  $ 1, 2, \ldots, 52$ . Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let  $ p(a)$  be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards  $ a$  and  $ a\plus{}9$ , and Dylan picks the other of these two cards. The minimum value of  $ p(a)$  for which  $ p(a)\ge\frac12$  can be written as  $ \frac{m}{n}$ . where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m\plus{}n$ .

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