
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 $60$ cards numbered $1, 2, \cdots, 60$, Alex, Blair, Corey, and Dylan each pick $2$ cards from the deck, sequentially without replacement, with each card being equally likely to be picked. Each person's lowest card determines their initial position. The two persons with the lowest two initial cards form one team, and the two with the highest initial cards form another team. Let $p(a,b)$ be the probability that Alex and Dylan are on the same team, given that Alex picks a set of cards containing $a$ and $b$ (where $b = a+10$), and Dylan picks from the remaining cards. Determine the minimum values of $a$ for which $p(a,b) \geq \frac{1}{2}$.

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