Update to ACK runtime v0.44.0, code-generator v0.44.0 (aws-controllers-k8s#28)

### Update to ACK runtime `v0.44.0`, code-generator `v0.44.0`

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* ACK code-generator `v0.44.0` [release notes](https://github.com/aws-controllers-k8s/code-generator/releases/tag/v0.44.0)
* ACK runtime `v0.44.0` [release notes](https://github.com/aws-controllers-k8s/runtime/releases/tag/v0.44.0)

----------

NOTE:
This PR increments the release version of service controller from `v1.0.5` to `v1.0.6`

Once this PR is merged, release `v1.0.6` will be automatically created for `apigateway-controller`

**Please close this PR, if you do not want the new patch release for `apigateway-controller`**

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#### stdout for `make build-controller`:

```
building ack-generate ... ok.
==== building apigateway-controller ====
Copying common custom resource definitions into apigateway
Building Kubernetes API objects for apigateway
Generating deepcopy code for apigateway
Generating custom resource definitions for apigateway
Building service controller for apigateway
Running GO mod tidy
Generating RBAC manifests for apigateway
Running gofmt against generated code for apigateway
Updating additional GitHub repository maintenance files
==== building apigateway-controller release artifacts ====
Building release artifacts for apigateway-v1.0.6
Generating common custom resource definitions
Generating custom resource definitions for apigateway
Generating RBAC manifests for apigateway
```

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By submitting this pull request, I confirm that my contribution is made under the terms of the Apache 2.0 license.

Author: ack-bot

apis/v1alpha1/ack-generate-metadata.yaml

@@ -1,13 +1,13 @@
 ack_generate_info:
-  build_date: "2025-02-20T18:04:23Z"
-  build_hash: a326346bd3a6973254d247c9ab2dc76790c36241
+  build_date: "2025-04-03T05:54:37Z"
+  build_hash: cccb1805b425c5a5e2bb8b99e22b0a811e2896e2
   go_version: go1.24.0
-  version: v0.43.2
-api_directory_checksum: 2b29d9314a6ca77238f1a5fe09772f307485d94f
+  version: v0.43.2-12-gcccb180
+api_directory_checksum: b36e0217b62cb5431d11ccfaad66312c1cf1aa35
 api_version: v1alpha1
-aws_sdk_go_version: 1.32.6
+aws_sdk_go_version: v1.32.6
 generator_config_info:
-  file_checksum: d2f4a656f9dccd7026fa8ab2be568da2b62379f6
+  file_checksum: 65d70046b645ac8c066d287a3c17fecc74270161
   original_file_name: generator.yaml
 last_modification:
   reason: API generation

apis/v1alpha1/api_method_response.go

@@ -5,6 +5,21 @@ operations:
     operation_type:
       - Create
     resource_name: Integration
+  PutMethodResponse:
+    operation_type:
+      - Create
+    resource_name: ApiMethodResponse
+  DeleteMethodResponse:
+    operation_type:
+      - Delete
+    resource_name: ApiMethodResponse
+  UpdateMethodResponse:
+    operation_type:
+      - Update
+    resource_name: ApiMethodResponse
+  GetMethodResponse:
+    operation_type:
+      - READ_ONE
 ignore:
   resource_names:
     - ApiKey
@@ -200,3 +215,36 @@ resources:
         - ConflictException
         - NotFoundException
         - InvalidParameter
+  ApiMethodResponse:
+    fields:
+      RestAPIID:
+        references:
+          resource: RestAPI
+          path: Status.ID
+        is_required: true
+        is_immutable: true
+      ResourceID:
+        references:
+          resource: Resource
+          path: Status.ID
+        is_required: true
+        is_immutable: true
+      HTTPMethod:
+        is_required: true
+        is_immutable: true
+      StatusCode:
+        is_required: true
+        is_immutable: true
+    tags:
+      ignore: true
+    hooks:
+      sdk_update_post_build_request:
+        template_path: hooks/api_method_response/sdk_update_post_build_request.go.tpl
+      delta_pre_compare:
+        code: customPreCompare(a, b)
+    exceptions:
+      terminal_codes:
+        - BadRequestException
+        - ConflictException
+        - NotFoundException
+        - InvalidParameter