```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);
"""

_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>

__device__ __forceinline__ float max_nan(float a, float b) {
    return (isnan(a) || isnan(b)) ? CUDART_NAN_F : fmaxf(a, b);
}

__device__ __forceinline__ float warp_sum(float value) {
    #pragma unroll
    for (int offset = 16; offset > 0; offset >>= 1) {
        value += __shfl_down_sync(0xffffffff, value, offset);
    }
    return value;
}

template<bool PAIRS>
__global__ void pool_sum_scale_kernel(
    const float* __restrict__ input,
    float* __restrict__ output,
    int64_t columns,
    int64_t kernel_size,
    float scale
) {
    const int tid = threadIdx.x;
    const int64_t row = blockIdx.x;
    const float* row_input = input + row * columns;
    const int64_t pooled_columns = columns / kernel_size;

    float sum = 0.0f;

    if (PAIRS) {
        const float2* pairs = reinterpret_cast<const float2*>(row_input);
        for (int64_t i = tid; i < pooled_columns; i += blockDim.x) {
            const float2 values = pairs[i];
            sum += max_nan(values.x, values.y);
        }
    } else {
        for (int64_t i = tid; i < pooled_columns; i += blockDim.x) {
            const float* window = row_input + i * kernel_size;
            float maximum = -CUDART_INF_F;
            for (int64_t j = 0; j < kernel_size; ++j) {
                maximum = max_nan(maximum, window[j]);
            }
            sum += maximum;
        }
    }

    sum = warp_sum(sum);

    __shared__ float warp_sums[8];
    const int lane = tid & 31;
    const int warp = tid >> 5;

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

    if (warp == 0) {
        sum = lane < 8 ? warp_sums[lane] : 0.0f;
        sum = warp_sum(sum);
        if (lane == 0) {
            output[row] = sum * scale;
        }
    }
}

torch::Tensor pool_sum_scale_cuda(
    torch::Tensor x, int64_t kernel_size, double scale
) {
    TORCH_CHECK(x.is_cuda(), "x must be a CUDA tensor");
    TORCH_CHECK(x.scalar_type() == at::kFloat, "x must be float32");
    TORCH_CHECK(x.dim() == 2 && x.is_contiguous(),
                "x must be a contiguous matrix");
    TORCH_CHECK(kernel_size > 0 && x.size(1) >= kernel_size,
                "invalid pooling kernel size");

    const c10::cuda::CUDAGuard device_guard(x.device());
    auto output = torch::empty({x.size(0)}, x.options());

    if (x.size(0) == 0) {
        return output;
    }

    const auto stream = at::cuda::getCurrentCUDAStream();
    const dim3 blocks(static_cast<unsigned int>(x.size(0)));
    constexpr int threads = 256;

    if (kernel_size == 2 && x.size(1) % 2 == 0) {
        pool_sum_scale_kernel<true><<<blocks, threads, 0, stream>>>(
            x.data_ptr<float>(), output.data_ptr<float>(),
            x.size(1), kernel_size, static_cast<float>(scale));
    } else {
        pool_sum_scale_kernel<false><<<blocks, threads, 0, stream>>>(
            x.data_ptr<float>(), output.data_ptr<float>(),
            x.size(1), kernel_size, static_cast<float>(scale));
    }

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return output;
}
"""

_fused_ops = load_inline(
    name="fused_maxpool_sum_scale_cuda_v1",
    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 (
            x.is_cuda
            and x.dtype == torch.float32
            and x.ndim == 2
            and x.is_contiguous()
            and not (torch.is_grad_enabled() and x.requires_grad)
        ):
            return _fused_ops.pool_sum_scale_cuda(
                x, self.kernel_size, self.scale_factor
            )

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