
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 deck of $52$ cards are numbered from $1$ to $52$. Alex, Blair, Corey, and Dylan each picks a card. The two persons with the lower numbered cards form one team and those 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 a card numbered $a$ and Dylan picks a card numbered $a+k$ with $1 \leq k \leq 12$. Assume $a+k \leq 52$. The minimum value of $p(a)$, for which $p(a) \ge \frac{1}{2}$, can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.

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