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

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

torch::Tensor pool_sum_scale_cuda(
    torch::Tensor x, int64_t kernel_size, double scale_factor);
"""

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

template <typename scalar_t>
__global__ void pool_sum_scale_kernel(
    const scalar_t* __restrict__ x,
    scalar_t* __restrict__ output,
    int64_t width,
    int64_t kernel_size,
    int64_t pooled_width,
    double scale_factor) {
    using acc_t = at::acc_type<scalar_t, true>;
    __shared__ acc_t warp_sums[8];

    const int tid = threadIdx.x;
    const int lane = tid & 31;
    const int warp = tid >> 5;
    const scalar_t* row = x + static_cast<int64_t>(blockIdx.x) * width;

    acc_t sum = acc_t(0);

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

        for (int64_t j = 0; j < kernel_size; ++j) {
            const acc_t value = static_cast<acc_t>(row[start + j]);
            if (value > maximum || isnan(value)) {
                maximum = value;
            }
        }
        sum += maximum;
    }

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

    if (lane == 0) {
        warp_sums[warp] = sum;
    }
    __syncthreads();

    if (warp == 0) {
        sum = lane < 8 ? warp_sums[lane] : acc_t(0);

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

        if (lane == 0) {
            // Match the intermediate dtype of torch.sum before scaling.
            const scalar_t reduced = static_cast<scalar_t>(sum);
            output[blockIdx.x] = static_cast<scalar_t>(
                static_cast<acc_t>(reduced) *
                static_cast<acc_t>(scale_factor));
        }
    }
}

torch::Tensor pool_sum_scale_cuda(
    torch::Tensor x, int64_t kernel_size, double scale_factor) {
    TORCH_CHECK(x.is_cuda(), "x must be a CUDA tensor");
    TORCH_CHECK(x.dim() == 2, "x must be two-dimensional");
    TORCH_CHECK(kernel_size > 0, "kernel_size must be positive");
    TORCH_CHECK(x.size(1) >= kernel_size, "pooling output would be empty");

    const c10::cuda::CUDAGuard device_guard(x.device());
    x = x.contiguous();

    auto output = torch::empty({x.size(0)}, x.options());
    if (x.size(0) == 0) {
        return output;
    }

    const int64_t pooled_width = x.size(1) / kernel_size;
    const auto stream = at::cuda::getCurrentCUDAStream();

    AT_DISPATCH_FLOATING_TYPES_AND2(
        at::ScalarType::Half,
        at::ScalarType::BFloat16,
        x.scalar_type(),
        "pool_sum_scale_cuda",
        [&] {
            pool_sum_scale_kernel<scalar_t>
                <<<static_cast<unsigned int>(x.size(0)), 256, 0, stream>>>(
                    x.data_ptr<scalar_t>(),
                    output.data_ptr<scalar_t>(),
                    x.size(1),
                    kernel_size,
                    pooled_width,
                    scale_factor);
        });

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return output;
}
"""

_fused_ops = load_inline(
    name="fused_maxpool_sum_scale_cuda",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["pool_sum_scale_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) else kernel_size
        )
        self.scale_factor = scale_factor

    def forward(self, x):
        x = self.matmul(x)
        if not x.is_cuda or (torch.is_grad_enabled() and x.requires_grad):
            x = self.max_pool(x.unsqueeze(1)).squeeze(1)
            return x.sum(dim=1) * self.scale_factor
        return _fused_ops.pool_sum_scale_cuda(
            x, self.kernel_size, self.scale_factor
        )
```