In the land of Amestris, alchemists are known for bending matter to their will, shaping stone, fire, and metal in the name of war or progress. But among them stands one who chose a different path — Rutherford, the Nonmetal Alchemist. Unlike others, he turned his back on combat and committed his life to the pursuit of knowledge.

Lately, usage of transmutation across Amestris has grown in intensity, leading to the appearance of never-before-seen elements at an alarming rate! These elements possess unknown properties, and Rutherford believes that their electron configurations, more specifically, the number of electrons in the final shell of their atoms, hold the key to understanding them.

Every element in the universe is made of atoms, and each atom in its neutral state has a specific number of electrons equal to its atomic number $$$Z$$$. These electrons orbit the nucleus, and each electron is uniquely described by four integers called quantum numbers.

• The principal quantum number $$$n$$$: Each value of $$$n$$$ uniquely describes a shell, where $$$n \ge 1$$$
• The azimuthal quantum number $$$\ell$$$: Each $$$(n, \ell)$$$ pair uniquely describes a subshell, where $$$0 \le \ell \lt n$$$
• The magnetic quantum number $$$m$$$: Each $$$(n, \ell, m)$$$ triplet uniquely describes an orbital, where $$$-\ell \le m \le \ell$$$
• The spin quantum number $$$s$$$: Each $$$(n, \ell, m, s)$$$ tuple uniquely describes an electron, where $$$1 \le s \le S$$$

The number of electrons that can fill in a subshell $$$(n, \ell)$$$ is determined by the number of valid $$$(n, \ell, m, s)$$$ combinations (each representing an electron). For example, if $$$S = 2$$$, the subshell $$$(2, 1)$$$ has a capacity of $$$6$$$ electrons. The quantum numbers for those $$$6$$$ electrons are $$$(2, 1, -1, 1)$$$, $$$(2, 1, -1, 2)$$$, $$$(2, 1, 0, 1)$$$, $$$(2, 1, 0, 2)$$$, $$$(2, 1, 1, 1)$$$, $$$(2, 1, 1, 2)$$$.

For any atom, electrons fill subshells in increasing order of energy, following the Aufbau principle. The energy of a subshell is primarily determined by $$$n + \ell$$$, and in case of ties, the subshell with the smaller $$$n$$$ is filled first. According to this rule, the first $$$10$$$ subshells filled are: $$$(1, 0)$$$, $$$(2, 0)$$$, $$$(2, 1)$$$, $$$(3, 0)$$$, $$$(3, 1)$$$, $$$(4, 0)$$$, $$$(3, 2)$$$, $$$(4, 1)$$$, $$$(5, 0)$$$, $$$(4, 2)$$$. The final shell of an atom is the largest value of $$$n$$$ that has a non-zero number of electrons.

For $$$S = 2$$$, the electrons of Germanium, an atom with atomic number $$$32$$$, enter the subshells in the following order:

• First, $$$2$$$ electrons enter the subshell $$$(1, 0)$$$, completely filling it.
• Then, $$$2$$$ electrons enter the subshell $$$(2, 0)$$$, completely filling it.
• Then, $$$6$$$ electrons enter the subshell $$$(2, 1)$$$, completely filling it.
• Then, $$$2$$$ electrons enter the subshell $$$(3, 0)$$$, completely filling it.
• Then, $$$6$$$ electrons enter the subshell $$$(3, 1)$$$, completely filling it.
• Then, $$$2$$$ electrons enter the subshell $$$(4, 0)$$$, completely filling it.
• Then, $$$10$$$ electrons enter the subshell $$$(3, 2)$$$, completely filling it.
• Finally, the remaining $$$2$$$ electrons enter the subshell $$$(4, 1)$$$.

The electron configuration of Germanium can be written in shorthand as $$$(1, 0)^2 (2, 0)^2 (2, 1)^6 (3, 0)^2 (3, 1)^6 (3, 2)^{10} (4, 0)^2 (4, 1)^2$$$, where $$$(n, \ell)^k$$$ means that $$$k$$$ electrons are placed in the subshell $$$(n, \ell)$$$. So, the final shell for this atom is shell $$$4$$$, and the number of electrons on the final shell is $$$2 + 2 = 4$$$.

You are Bohr, Rutherford's student — curious, determined, and eager to assist your mentor in his mission. Help Rutherford calculate the number of electrons in the final shell for any atom, given its atomic number $$$Z$$$ and $$$S$$$, the number of possible spin values for electrons in this universe.