The image above shows two circular planar portals: a blue portal, with its axis oriented in the vertical direction, and an orange portal, with its axis oriented in the horizontal direction. The blue portal is situated at \(y=0\), and its \(x\)-coordinate is irrelevant. The orange portal is situated at \(x=0\), and its height (\(y\)-position) varies as described later.

The rules of portal physics are the following:

- An object enters one portal and emerges from the other
- Portals
**do not**conserve vector momentum or total energy (kinetic + potential) - Portals
**do**conserve the scalar speed of objects that enter / exit them - An object entering one portal at its center on a trajectory perpendicular to its axis, will emerge from the other portal in like manner
- Portals do not impart their own velocity to objects which pass through them
- Aside from their own strange properties, portals do not otherwise alter the physics of nearby objects and environments
- It takes zero time to go through a pair of portals

At time \(t = 0\), two things happen simultaneously:

A massive ball drops from its initial resting position at height \(h_B\) and falls under the influence of a uniform downward gravitational acceleration \(g\). It falls toward the center of the blue portal.

The orange portal (position defined by its center) begins to descend from its initial vertical position \(h_P\) at a constant speed \(v_P\).

Consider the \(x\)-coordinate of the ball at the instant at which it intersects the \(x\)-axis after emerging from the orange portal. The value of \(h_B\) which maximizes this quantity can expressed as:

\[{h_{B_\text{max}} = \frac{a}{b} g \left(\frac{h_P}{v_P}\right)^2}.\]

If \(a\) and \(b\) are coprime positive integers, determine the value of \(a+b\).

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