Accurate reconstruction of arbitrary-shaped long slender continuum bodies, such as guidewires, catheters and other soft continuum manipulators, is essential for accurate mechanical simulation. However, existing image-based reconstruction approaches often suffer from limited accuracy because they often underutilize camera geometry, or lack generality as they rely on rigid geometric assumptions that may fail for continuum robots with complex and highly deformable shapes. To address these limitations, we propose ACCURATE, a 3D reconstruction framework integrating an image segmentation neural network with a geometry-constrained topology traversal and dynamic programming algorithm that enforces global biplanar geometric consistency, minimizes the cumulative point-to-epipolar-line distance, and remains robust to occlusions and epipolar ambiguities cases caused by noise and discretization. Our method achieves high reconstruction accuracy on both simulated and real phantom datasets acquired using a clinical X-ray C-arm system, with mean absolute errors below 1.0 mm.