
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 a rhombus with side length 7 and angles at vertices $A$ and $C$ equal to $60^\circ$. Circles are drawn with radii 4 centered at $A$ and $D$, and radii 3 centered at $B$ and $C$. Find the radius of another circle that lies within the rhombus and is tangent to all four of these circles. Express the radius as $\frac{a + b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are integers, $c$ is not divisible by the square of any prime, and $a$ and $d$ are relatively prime. Find $a+b+c+d$.

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