feat(ci): update test workflow to use go.mod version file and verify build

Author: Jing-yilin

.github/workflows/test-backend.yml

@@ -4,9 +4,11 @@ on:
   push:
     paths:
       - "backend/**"
+      - ".github/workflows/test-backend.yml"
   pull_request:
     paths:
       - "backend/**"
+      - ".github/workflows/test-backend.yml"
 
 jobs:
   test:
@@ -18,8 +20,8 @@ jobs:
       - uses: actions/checkout@v4
       - uses: actions/setup-go@v5
         with:
-          go-version: "1.24"
-          cache-dependency-path: backend/go.sum
-      - run: go build ./...
-      - run: go test ./...
+          go-version-file: go.mod
+          cache-dependency-path: go.sum
       - run: go vet ./...
+      - run: go test ./...
+      - run: CGO_ENABLED=0 GOOS=linux go build -o /dev/null ./cmd/api