
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
A deck of $52$ cards is numbered $1$ to $52$. Four friends, Ella, George, Lee, and Zoe, each randomly pick a card, such that no replacement occurs and the choice of each card is equally likely. The participants with the lower two card numbers form one team, and those with the higher two numbers form the other team. Let $p(b)$ be the probability that Ella and Zoe are on the same team, given that Ella chooses card $b$ or $b+11$, and Zoe picks the other of these two cards. Determine the minimum value of $b$ such that $p(b) \geq \frac{1}{2}$ and express $p(b)$ in the form of a simplified fraction $\frac{m}{n}$ where $m$ and $n$ are coprime numbers. Find the sum $m+n$.

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