A company has three factories, each manufacturing the same product. Let x, y, and z represent the number of units produced by factory A, B, and C, respectively, with the corresponding manufacturing cost of 3x² + 200, +400, and 2z² + 300 pesos If an order for 121,000 units is to be filled, use Lagrange multipliers to determine how the production should be distributed among the three factories to minimize the total manufacturing cost.

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