[examples] Update APIGateway example with the new APIGatewayResponse initializer #362

Job	Run time
musl	1m 28s
Semantic Version label check	3s
Unit tests / Construct unit test matrix	4s
Soundness / API breakage check	1m 39s
Soundness / Broken symlinks check	5s
Soundness / Documentation check	1m 25s
Soundness / Format check	32s
Soundness / License headers check	4s
Soundness / Python lint check	9s
Soundness / Shell check	36s
Soundness / Unacceptable language check	3s
Soundness / YAML lint check	3s
Integration Tests / Test archive plugin	2m 35s
Integration Tests / Test Examples/APIGateway on 6.0.1-amazonlinux2	1m 20s
Integration Tests / Test Examples/BackgroundTasks on 6.0.1-amazonlinux2	1m 14s
Integration Tests / Test Examples/HelloJSON on 6.0.1-amazonlinux2	1m 13s
Integration Tests / Test Examples/HelloWorld on 6.0.1-amazonlinux2	1m 15s
Integration Tests / Test Examples/S3_AWSSDK on 6.0.1-amazonlinux2	4m 7s
Integration Tests / Test Examples/S3_Soto on 6.0.1-amazonlinux2	4m 33s
Integration Tests / Test Examples/Streaming on 6.0.1-amazonlinux2	1m 18s
Swift 6 Language Mode / Swift 6 language mode	1m 10s
Unit tests / Unit tests / Linux (6.0)	1m 17s
Unit tests / Unit tests / Linux (nightly-6.0)	1m 32s
Unit tests / Unit tests / Linux (nightly-main)	1m 31s
	29m 16s

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