
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 $40$ cards, numbered $1$ to $40$, is used. Carly, Drew, Erin, and Fiona each pick one card from the deck without replacement, with each card equally likely to be picked. The two persons with lower numbered cards form one team, and those with higher numbered cards form another team. Let $q(b)$ be the probability that Carly and Fiona are on the same team when Carly picks card number $b$ and Fiona picks card number $b+7$. Determine the smallest value of $b$ for which $q(b) \ge \frac{1}{2}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find the sum $p+q$.

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