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

_cpp_source = """
torch::Tensor pool_sum_scale_cuda(
    torch::Tensor x, int64_t kernel_size, double scale);
"""

_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_runtime.h>

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

template <typename scalar_t, typename acc_t, int FIXED_K>
__global__ void pool_sum_scale_kernel(
    const scalar_t* __restrict__ x,
    scalar_t* __restrict__ out,
    int64_t width,
    int64_t runtime_k,
    acc_t scale)
{
    const int64_t k = FIXED_K ? FIXED_K : runtime_k;
    const int64_t pooled_width = width / k;
    const scalar_t* row = x + static_cast<int64_t>(blockIdx.x) * width;

    acc_t total = acc_t(0);
    for (int64_t p = threadIdx.x; p < pooled_width; p += blockDim.x) {
        const int64_t start = p * k;
        acc_t maximum = static_cast<acc_t>(row[start]);

        for (int64_t j = 1; j < k; ++j) {
            acc_t value = static_cast<acc_t>(row[start + j]);
            if (value > maximum || isnan(value))
                maximum = value;
        }
        total += maximum;
    }

    total = warp_sum(total);

    __shared__ acc_t warp_totals[8];
    const int lane = threadIdx.x & 31;
    const int warp = threadIdx.x >> 5;

    if (lane == 0)
        warp_totals[warp] = total;
    __syncthreads();

    if (warp == 0) {
        total = lane < 8 ? warp_totals[lane] : acc_t(0);
        total = warp_sum(total);

        if (lane == 0) {
            // Preserve the intermediate rounding performed by torch.sum.
            scalar_t reduced = static_cast<scalar_t>(total);
            out[blockIdx.x] = static_cast<scalar_t>(
                static_cast<acc_t>(reduced) * scale);
        }
    }
}

torch::Tensor pool_sum_scale_cuda(
    torch::Tensor x, int64_t kernel_size, double scale)
{
    TORCH_CHECK(x.is_cuda(), "Expected a CUDA tensor");
    TORCH_CHECK(x.dim() == 2 && x.is_contiguous(),
                "Expected a contiguous two-dimensional tensor");
    TORCH_CHECK(kernel_size > 0 && kernel_size <= x.size(1),
                "Invalid pooling kernel size");

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

    const auto stream = at::cuda::getCurrentCUDAStream();

    AT_DISPATCH_FLOATING_TYPES_AND2(
        at::ScalarType::Half,
        at::ScalarType::BFloat16,
        x.scalar_type(),
        "pool_sum_scale_cuda",
        [&] {
            using acc_t = at::acc_type<scalar_t, true>;
            if (kernel_size == 2) {
                pool_sum_scale_kernel<scalar_t, acc_t, 2>
                    <<<x.size(0), 256, 0, stream>>>(
                        x.data_ptr<scalar_t>(),
                        out.data_ptr<scalar_t>(),
                        x.size(1), kernel_size,
                        static_cast<acc_t>(scale));
            } else {
                pool_sum_scale_kernel<scalar_t, acc_t, 0>
                    <<<x.size(0), 256, 0, stream>>>(
                        x.data_ptr<scalar_t>(),
                        out.data_ptr<scalar_t>(),
                        x.size(1), kernel_size,
                        static_cast<acc_t>(scale));
            }
        });

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return out;
}
"""

_fused_ops = load_inline(
    name="linear_pool_sum_scale_fused_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.kernel_size = 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 = F.max_pool1d(x.unsqueeze(1), self.kernel_size).squeeze(1)
            return x.sum(dim=1) * self.scale_factor

        return _fused_ops.pool_sum_scale_cuda(
            x.contiguous(), self.kernel_size, self.scale_factor
        )
```