
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 deck of 52 cards numbered 1 to 52, Alex, Blair, Corey, and Dylan each pick a card without replacement, each card being equally likely to be picked. The two persons with lower numbered cards form a team, and the two 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 \( a+15 \), and Dylan picks the other of these two cards. Find the minimum value of \( p(a) \) such that \( p(a) \geq \frac{1}{2} \).

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