
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 $60$ cards in a deck are numbered $1, 2, \cdots, 60$. Alice, Bob, Charlie, and Dave 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 form a team, and the two persons with higher numbered cards form another team. Let $q(a)$ be the probability that Alice and Dave are on the same team, given that Alice picks one of the cards $a$ and $a+11$, and Dave picks the other of these two cards. The minimum value of $q(a)$ for which $q(a)\ge\frac{1}{2}$ can be written as $\frac{p}{q}$. where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

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