Two Types of Spells

Polycarp plays a computer game (yet again). In this game, he fights monsters using magic spells.

There are two types of spells: fire spell of power $$$x$$$ deals $$$x$$$ damage to the monster, and lightning spell of power $$$y$$$ deals $$$y$$$ damage to the monster and doubles the damage of the next spell Polycarp casts. Each spell can be cast only once per battle, but Polycarp can cast them in any order.

For example, suppose that Polycarp knows three spells: a fire spell of power $$$5$$$, a lightning spell of power $$$1$$$, and a lightning spell of power $$$8$$$. There are $$$6$$$ ways to choose the order in which he casts the spells:

• first, second, third. This order deals $$$5 + 1 + 2 \cdot 8 = 22$$$ damage;
• first, third, second. This order deals $$$5 + 8 + 2 \cdot 1 = 15$$$ damage;
• second, first, third. This order deals $$$1 + 2 \cdot 5 + 8 = 19$$$ damage;
• second, third, first. This order deals $$$1 + 2 \cdot 8 + 2 \cdot 5 = 27$$$ damage;
• third, first, second. This order deals $$$8 + 2 \cdot 5 + 1 = 19$$$ damage;
• third, second, first. This order deals $$$8 + 2 \cdot 1 + 2 \cdot 5 = 20$$$ damage.

Initially, Polycarp knows $$$0$$$ spells. His spell set changes $$$n$$$ times, each time he either learns a new spell or forgets an already known one. After each change, calculate the maximum possible damage Polycarp may deal using the spells he knows.