
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 from $1$ to $52$. Charlie, Sam, Lisa, and Jane each draw a card from the deck without replacement, ensuring that each card has an equal likelihood of being drawn. The two individuals with smaller numbered cards form one team, and the two with larger numbered cards form another team. Define $p(a)$ as the probability that Charlie and Jane are on the same team, given that Charlie picks one of the cards $a$ and $a+11$, and Jane picks the other of these two cards. The minimum value of $p(a)$ for which $p(a)$ is at least $\frac{1}{2}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are coprime integers. Find the sum $m+n$.

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