Allow full web search tool config (#13675)

Previously, we could only configure whether web search was on/off.

This PR enables sending along a web search config, which includes all
the stuff responsesapi supports: filters, location, etc.

codex-rs/app-server-protocol/schema/json/v2/ConfigReadResponse.json

@@ -628,6 +628,16 @@
             }
           ]
         },
+        "tools": {
+          "anyOf": [
+            {
+              "$ref": "#/definitions/ToolsV2"
+            },
+            {
+              "type": "null"
+            }
+          ]
+        },
         "web_search": {
           "anyOf": [
             {
@@ -721,9 +731,13 @@
           ]
         },
         "web_search": {
-          "type": [
-            "boolean",
-            "null"
+          "anyOf": [
+            {
+              "$ref": "#/definitions/WebSearchToolConfig"
+            },
+            {
+              "type": "null"
+            }
           ]
         }
       },
@@ -738,6 +752,44 @@
       ],
       "type": "string"
     },
+    "WebSearchContextSize": {
+      "enum": [
+        "low",
+        "medium",
+        "high"
+      ],
+      "type": "string"
+    },
+    "WebSearchLocation": {
+      "additionalProperties": false,
+      "properties": {
+        "city": {
+          "type": [
+            "string",
+            "null"
+          ]
+        },
+        "country": {
+          "type": [
+            "string",
+            "null"
+          ]
+        },
+        "region": {
+          "type": [
+            "string",
+            "null"
+          ]
+        },
+        "timezone": {
+          "type": [
+            "string",
+            "null"
+          ]
+        }
+      },
+      "type": "object"
+    },
     "WebSearchMode": {
       "enum": [
         "disabled",
@@ -745,6 +797,41 @@
         "live"
       ],
       "type": "string"
+    },
+    "WebSearchToolConfig": {
+      "additionalProperties": false,
+      "properties": {
+        "allowed_domains": {
+          "items": {
+            "type": "string"
+          },
+          "type": [
+            "array",
+            "null"
+          ]
+        },
+        "context_size": {
+          "anyOf": [
+            {
+              "$ref": "#/definitions/WebSearchContextSize"
+            },
+            {
+              "type": "null"
+            }
+          ]
+        },
+        "location": {
+          "anyOf": [
+            {
+              "$ref": "#/definitions/WebSearchLocation"
+            },
+            {
+              "type": "null"
+            }
+          ]
+        }
+      },
+      "type": "object"
     }
   },
   "properties": {