Pay-After-You-Play Amusement Park and the Digital Escape (AI)

Problem Statement

"Light is not only a wave; it also possesses the properties of discrete packets of energy (particles)."

This concept, proposed by Albert Einstein, is known as the "light-quantum hypothesis." Light behaves as a collection of particles, each carrying an energy defined by $E = hf$E=hf (where $h$h is Planck's constant and $f$f is the frequency of the light).

When light with sufficient energy strikes the surface of a metal, the electrons within the metal absorb the energy from these light particles in their entirety and are ejected from the surface. This phenomenon is called the "photoelectric effect."

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The process of an electron being ejected from the surface of a metal can be likened to a game: "Escaping from an amusement park that uses a 'pay-upon-exit' system."

1. Exit Fee (Work Function, $W$W) For a "guest" (an electron) inside the "amusement park" (the metal) to escape, they must pay an "exit fee"—an amount of energy (or money) equal to at least $W$W—at the checkout counter near the exit.
2. Found Banknote (Energy of a Single Light Particle, $E = hf$E=hf) Scattered on the ground inside the park are "banknotes"—light particles that have fallen from above—each with a value of $E = hf$E=hf. Each guest can pick up only one banknote; they cannot share a banknote with others, nor can they split it into smaller amounts.
3. Remaining Funds upon Exit (Kinetic Energy, $K$K)

• If the value of the banknote ($E$E) is less than the required exit fee ($W$W)—that is, $E < W$E<W—the guest cannot afford the fee and, therefore, cannot escape the park. * If the value $E$E of a banknote is greater than or equal to the admission fee $W$W ($E \ge W$E≥W), a visitor can pay the fee $W$W at the exit counter to leave the park, with the remaining "change" ($E - W$E−W) becoming their "exit momentum" (kinetic energy $K$K).

In this scenario, the kinetic energy $K$K (the change) of a visitor exiting the park is expressed by Einstein’s photoelectric equation:

$K = E - W = hf - W$K=E−W=hf−W

Suppose a visitor inside the park finds a banknote with a value of $E_1 = hf_1$E1​=hf1​ (representing a light particle with frequency $f_1$f1​) and uses it to pay the admission fee; the visitor then exits the park with an amount of "change" (kinetic energy) equal to $K_1$K1​.

Next, if the value of the banknote found is increased fourfold to $4E_1 = h(4f_1)$4E1​=h(4f1​) (representing a light particle with a frequency four times greater), and a visitor uses this to pay the admission fee, the "change" (kinetic energy) they possess upon exiting becomes $K_2 = 13K_1$K2​=13K1​ (thirteen times the initial amount).

Determine the ratio of the admission fee (work function $W$W) to the value of the first banknote found ($E_1 = hf_1$E1​=hf1​).

Constraints

• $f_1 > 0 \quad \mathrm{Hz}$f1​>0Hz
• $h > 0 \quad \mathrm{J \cdot s}$h>0J⋅s
• $W > 0 \quad \mathrm{J}$W>0J

Input Format

The ratio $\dfrac{W}{E_1}$E1​W​ can be expressed as an irreducible fraction $\dfrac{p}{q}$qp​ (where $p$p and $q$q are coprime natural numbers). Provide the value of $p \times q$p×q.