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The GeGLU (GELU-based Gated Linear Unit) activation function is defined by the following formula: $$\sigma_{\text{geglu}}(\mathbf{h}) = \sigma_{\text{gelu}}(\mathbf{hW}_1 + \mathbf{b}_1) \odot (\mathbf{hW}_2 + \mathbf{b}_2)$$ In this equation, $$\mathbf{h}$$ represents the input, while $$\mathbf{W}_1$$, $$\mathbf{W}_2$$, $$\mathbf{b}_1$$, and $$\mathbf{b}_2$$ are learnable model parameters (weights and biases). The function $$\sigma_{\text{gelu}}$$ is the Gaussian Error Linear Unit (GELU) activation, and $$\odot$$ signifies the element-wise product.

GeGLU (GELU-based Gated Linear Unit) Formula

You are debugging a neural network layer that uses the GeGLU activation function, defined as: `GeGLU(h) = GELU(hW₁ + b₁) ⊙ (hW₂ + b₂)`, where `⊙` represents the element-wise product. Given the case details below, calculate the final output value. Show the results of the two intermediate linear transformations before the final calculation.

GeGLU Activation Calculation

In the GeGLU activation function, defined as `σ_geglu(h) = σ_gelu(hW₁ + b₁) ⊙ (hW₂ + b₂)`, both of the linear transformations `(hW₁ + b₁)` and `(hW₂ + b₂)` are passed through the GELU activation function before the element-wise product is computed.

An activation function is defined by the formula: `f(x) = GELU(xW₁ + b₁) ⊙ (xW₂ + b₂)`, where `x` is the input, `W` and `b` are learnable parameters, `GELU` is an activation function, and `⊙` denotes an element-wise product. Based on this structure, what is the primary purpose of the `(xW₂ + b₂)` component?

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