```python
import torch
import torch.nn as nn
import torch.nn.functional as F
from torch.utils.cpp_extension import load_inline


_cpp_source = r"""
#include <torch/extension.h>

torch::Tensor pooled_sum_cuda(
    torch::Tensor input,
    torch::Tensor bias,
    int64_t kernel_size,
    double scale_factor);
"""

_cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <c10/cuda/CUDAGuard.h>
#include <c10/cuda/CUDAException.h>
#include <cuda_runtime.h>
#include <math.h>

template <int K>
__global__ void pooled_sum_kernel(
    const float* __restrict__ input,
    const float* __restrict__ bias,
    float* __restrict__ output,
    int64_t width,
    int64_t pooled_width,
    int64_t runtime_k,
    float scale)
{
    const int tid = threadIdx.x;
    const int lane = tid & 31;
    const int warp = tid >> 5;
    const int64_t k = K ? K : runtime_k;
    const float* row = input + static_cast<int64_t>(blockIdx.x) * width;

    float total = 0.0f;

    for (int64_t p = tid; p < pooled_width; p += blockDim.x) {
        const int64_t start = p * k;
        float maximum = -INFINITY;

        if (K == 2) {
            const float a = row[start] + bias[start];
            const float b = row[start + 1] + bias[start + 1];
            maximum = (isnan(a) || isnan(b)) ? NAN : fmaxf(a, b);
        } else {
            for (int64_t j = 0; j < k; ++j) {
                const float value = row[start + j] + bias[start + j];
                if (value > maximum || isnan(value)) {
                    maximum = value;
                }
            }
        }

        total += maximum;
    }

    #pragma unroll
    for (int offset = 16; offset > 0; offset >>= 1) {
        total += __shfl_down_sync(0xffffffff, total, offset);
    }

    __shared__ float warp_sums[8];
    if (lane == 0) {
        warp_sums[warp] = total;
    }
    __syncthreads();

    if (warp == 0) {
        total = lane < 8 ? warp_sums[lane] : 0.0f;

        #pragma unroll
        for (int offset = 16; offset > 0; offset >>= 1) {
            total += __shfl_down_sync(0xffffffff, total, offset);
        }

        if (lane == 0) {
            output[blockIdx.x] = total * scale;
        }
    }
}

torch::Tensor pooled_sum_cuda(
    torch::Tensor input,
    torch::Tensor bias,
    int64_t kernel_size,
    double scale_factor)
{
    TORCH_CHECK(input.is_cuda() && bias.is_cuda(),
                "input and bias must be CUDA tensors");
    TORCH_CHECK(input.device() == bias.device(),
                "input and bias must be on the same device");
    TORCH_CHECK(input.scalar_type() == at::kFloat &&
                bias.scalar_type() == at::kFloat,
                "input and bias must have dtype float32");
    TORCH_CHECK(input.dim() == 2 && bias.dim() == 1,
                "expected a matrix and a bias vector");
    TORCH_CHECK(input.is_contiguous() && bias.is_contiguous(),
                "input and bias must be contiguous");
    TORCH_CHECK(bias.numel() == input.size(1),
                "bias size does not match input width");
    TORCH_CHECK(kernel_size > 0 && kernel_size <= input.size(1),
                "invalid pooling kernel size");

    const c10::cuda::CUDAGuard device_guard(input.device());
    const int64_t batch = input.size(0);
    const int64_t width = input.size(1);
    auto output = torch::empty({batch}, input.options());

    if (batch == 0) {
        return output;
    }

    const auto stream = at::cuda::getCurrentCUDAStream();
    const dim3 grid(static_cast<unsigned int>(batch));

    if (kernel_size == 2) {
        pooled_sum_kernel<2><<<grid, 256, 0, stream>>>(
            input.data_ptr<float>(),
            bias.data_ptr<float>(),
            output.data_ptr<float>(),
            width,
            width / kernel_size,
            kernel_size,
            static_cast<float>(scale_factor));
    } else {
        pooled_sum_kernel<0><<<grid, 256, 0, stream>>>(
            input.data_ptr<float>(),
            bias.data_ptr<float>(),
            output.data_ptr<float>(),
            width,
            width / kernel_size,
            kernel_size,
            static_cast<float>(scale_factor));
    }

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return output;
}
"""

_fused_ops = load_inline(
    name="linear_bias_maxpool_sum_scale_cuda",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["pooled_sum_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(self, in_features, out_features, kernel_size, scale_factor):
        super().__init__()
        self.matmul = nn.Linear(in_features, out_features)
        self.max_pool = nn.MaxPool1d(kernel_size)
        self.kernel_size = (
            kernel_size[0]
            if isinstance(kernel_size, (tuple, list))
            else kernel_size
        )
        self.scale_factor = scale_factor

    def forward(self, x):
        if (
            x.is_cuda
            and x.dtype == torch.float32
            and self.matmul.weight.dtype == torch.float32
            and not torch.is_autocast_enabled()
        ):
            projected = F.linear(x, self.matmul.weight, bias=None)
            return _fused_ops.pooled_sum_cuda(
                projected,
                self.matmul.bias,
                self.kernel_size,
                self.scale_factor,
            )

        x = self.matmul(x)
        x = self.max_pool(x.unsqueeze(1)).squeeze(1)
        return torch.sum(x, dim=1) * self.scale_factor
```