Fix SDK path: add settingSources and default system prompt (#726)

Two fixes for the Agent SDK path (USE_AGENT_SDK=true):

1. Add settingSources to load filesystem settings
   - Without this, CLI-installed plugins aren't available to the SDK
   - Also needed to load CLAUDE.md files from the project

2. Default systemPrompt to claude_code preset
   - Without an explicit systemPrompt, the SDK would use no system prompt
   - Now defaults to { type: "preset", preset: "claude_code" } to match CLI behavior

Also adds logging of SDK options (excluding env) for debugging.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-authored-by: Claude <noreply@anthropic.com>

base-action/src/parse-sdk-options.ts

@@ -101,7 +101,7 @@ export function parseSdkOptions(options: ClaudeOptions): ParsedSdkOptions {
     env.GITHUB_ACTION_INPUTS = process.env.INPUT_ACTION_INPUTS_PRESENT;
   }
 
-  // Build system prompt option
+  // Build system prompt option - default to claude_code preset
   let systemPrompt: SdkOptions["systemPrompt"];
   if (options.systemPrompt) {
     systemPrompt = options.systemPrompt;
@@ -111,6 +111,12 @@ export function parseSdkOptions(options: ClaudeOptions): ParsedSdkOptions {
       preset: "claude_code",
       append: options.appendSystemPrompt,
     };
+  } else {
+    // Default to claude_code preset when no custom prompt is specified
+    systemPrompt = {
+      type: "preset",
+      preset: "claude_code",
+    };
   }
 
   // Build SDK options - use merged tools from both direct options and claudeArgs
@@ -130,6 +136,9 @@ export function parseSdkOptions(options: ClaudeOptions): ParsedSdkOptions {
     // Note: allowedTools and disallowedTools have been removed from extraArgs to prevent duplicates
     extraArgs,
     env,
+
+    // Load settings from all sources to pick up CLI-installed plugins, CLAUDE.md, etc.
+    settingSources: ["user", "project", "local"],
   };
 
   return {