```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_and_sum_cuda(torch::Tensor x);
"""

_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>
#include <cmath>

template <typename T>
__device__ __forceinline__ T maximum(T a, T b) {
    return isnan(b) || b > a ? b : a;
}

template <typename scalar_t>
__global__ void pool_and_sum_kernel(
    const scalar_t* __restrict__ x,
    scalar_t* __restrict__ out,
    int64_t channels,
    int64_t depth,
    int64_t height,
    int64_t width,
    int64_t out_depth,
    int64_t out_height,
    int64_t out_width)
{
    using acc_t = at::acc_type<scalar_t, true>;
    __shared__ acc_t warp_sums[8];

    const int lane = threadIdx.x & 31;
    const int warp = threadIdx.x >> 5;
    const int64_t output_index = blockIdx.x;

    int64_t position = output_index;
    const int64_t ow = position % out_width;
    position /= out_width;
    const int64_t oh = position % out_height;
    position /= out_height;
    const int64_t od = position % out_depth;
    const int64_t batch = position / out_depth;

    const int64_t spatial_size = depth * height * width;
    const int64_t start = (od * 6 * height + oh * 6) * width + ow * 6;
    acc_t channel_sum = acc_t(0);

    // Pooling with sizes/strides 2 and then 3 is exactly pooling
    // with size/stride 6. Each warp handles one channel at a time.
    for (int64_t channel = warp; channel < channels; channel += 8) {
        const scalar_t* base =
            x + (batch * channels + channel) * spatial_size + start;
        acc_t value = static_cast<acc_t>(-INFINITY);

        #pragma unroll
        for (int i = lane; i < 216; i += 32) {
            const int dz = i / 36;
            const int dy = (i / 6) % 6;
            const int dx = i % 6;
            const acc_t sample =
                static_cast<acc_t>(base[(dz * height + dy) * width + dx]);
            value = maximum(value, sample);
        }

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

        if (lane == 0) {
            channel_sum += value;
        }
    }

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

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

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

        if (lane == 0) {
            out[output_index] = static_cast<scalar_t>(value);
        }
    }
}

torch::Tensor pool_and_sum_cuda(torch::Tensor x) {
    TORCH_CHECK(x.is_cuda(), "Expected a CUDA tensor");
    TORCH_CHECK(x.dim() == 5, "Expected an NCDHW tensor");
    TORCH_CHECK(x.is_contiguous(), "Expected contiguous input");
    TORCH_CHECK(
        x.size(2) >= 6 && x.size(3) >= 6 && x.size(4) >= 6,
        "Spatial dimensions must be at least 6");

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

    const int64_t od = x.size(2) / 6;
    const int64_t oh = x.size(3) / 6;
    const int64_t ow = x.size(4) / 6;
    auto out = torch::empty({x.size(0), 1, od, oh, ow}, x.options());

    if (out.numel() == 0) {
        return out;
    }

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

    AT_DISPATCH_FLOATING_TYPES_AND2(
        at::ScalarType::Half,
        at::ScalarType::BFloat16,
        x.scalar_type(),
        "pool_and_sum_cuda",
        [&] {
            pool_and_sum_kernel<scalar_t>
                <<<static_cast<unsigned int>(out.numel()), 256, 0, stream>>>(
                    x.data_ptr<scalar_t>(),
                    out.data_ptr<scalar_t>(),
                    x.size(1),
                    x.size(2),
                    x.size(3),
                    x.size(4),
                    od, oh, ow);
        });

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return out;
}
"""

_fused_ops = load_inline(
    name="fused_maxpool3d_channel_sum",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["pool_and_sum_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(self, in_channels, out_channels, kernel_size, stride, padding):
        super().__init__()
        self.conv_transpose = nn.ConvTranspose3d(
            in_channels,
            out_channels,
            kernel_size,
            stride=stride,
            padding=padding,
        )

    def forward(self, x):
        x = self.conv_transpose(x)
        return _fused_ops.pool_and_sum_cuda(x.contiguous())
```