
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
Let $ABCD$ be an isosceles trapezoid with $AB = 8, BC = 7 = DA,$ and $CD = 6$. Circles of radius 4 are centered at $A$ and $B$, and circles of radius 3 are centered at $C$ and $D$. A circle inside the trapezoid is tangent to all four of these circles. Find the radius of this contained circle, expressed as $\frac{-k + m\sqrt{n}}{p}$, where $k, m, n, p$ are positive integers, $n$ is square-free, and $k$ and $p$ are relatively prime. Determine $k + m + n + p$.

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