The captivating maze-themed board games found in many toy stores are often used by school clubs to raise money for educational activities. A game piece, representing different denominations of money - pennies, nickels, dimes, and quarters - is moved along a winding path on the board, and it seemingly makes endless rounds before finally reaching the "treasure chest" (the score tally). After one such gaming session, the total amount represented by the game pieces reached $112.89, with 1450 pieces in play, each corresponding to a specific coin. How many game pieces corresponded to each coin, if the number of pieces representing quarters and dimes equaled the number of those for nickels, and the pieces representing pennies were twice the number of those for quarters? (Formulate a system of equations and solve it by utilizing the inverse matrix method.)