[Enhancement] Allow modify conversation variable via api (#23112)

Co-authored-by: autofix-ci[bot] <114827586+autofix-ci[bot]@users.noreply.github.com>

web/app/components/develop/template/template_advanced_chat.en.mdx

@@ -1011,6 +1011,121 @@ Chat applications support session persistence, allowing previous chat history to
 
 ---
 
+<Heading
+  url='/conversations/:conversation_id/variables/:variable_id'
+  method='PUT'
+  title='Update Conversation Variable'
+  name='#update-conversation-variable'
+/>
+<Row>
+  <Col>
+    Update the value of a specific conversation variable. This endpoint allows you to modify the value of a variable that was captured during the conversation while preserving its name, type, and description.
+
+    ### Path Parameters
+
+    <Properties>
+      <Property name='conversation_id' type='string' key='conversation_id'>
+        The ID of the conversation containing the variable to update.
+      </Property>
+      <Property name='variable_id' type='string' key='variable_id'>
+        The ID of the variable to update.
+      </Property>
+    </Properties>
+
+    ### Request Body
+
+    <Properties>
+      <Property name='value' type='any' key='value'>
+        The new value for the variable. Must match the variable's expected type (string, number, object, etc.).
+      </Property>
+      <Property name='user' type='string' key='user'>
+        The user identifier, defined by the developer, must ensure uniqueness within the application.
+      </Property>
+    </Properties>
+
+    ### Response
+
+    Returns the updated variable object with:
+    - `id` (string) Variable ID
+    - `name` (string) Variable name
+    - `value_type` (string) Variable type (string, number, object, etc.)
+    - `value` (any) Updated variable value
+    - `description` (string) Variable description
+    - `created_at` (int) Creation timestamp
+    - `updated_at` (int) Last update timestamp
+
+    ### Errors
+    - 400, `Type mismatch: variable expects {expected_type}, but got {actual_type} type`, Value type doesn't match variable's expected type
+    - 404, `conversation_not_exists`, Conversation not found
+    - 404, `conversation_variable_not_exists`, Variable not found
+
+  </Col>
+  <Col sticky>
+
+    <CodeGroup title="Request" tag="PUT" label="/conversations/:conversation_id/variables/:variable_id" targetCode={`curl -X PUT '${props.appDetail.api_base_url}/conversations/{conversation_id}/variables/{variable_id}' \\\n--header 'Authorization: Bearer {api_key}' \\\n--header 'Content-Type: application/json' \\\n--data-raw '{\n  "value": "Updated Value",\n  "user": "abc-123"\n}'`}>
+
+    ```bash {{ title: 'cURL' }}
+    curl -X PUT '${props.appDetail.api_base_url}/conversations/{conversation_id}/variables/{variable_id}' \
+    --header 'Content-Type: application/json' \
+    --header 'Authorization: Bearer {api_key}' \
+    --data-raw '{
+        "value": "Updated Value",
+        "user": "abc-123"
+    }'
+    ```
+
+    </CodeGroup>
+
+    <CodeGroup title="Update with different value types">
+    ```bash {{ title: 'String Value' }}
+    curl -X PUT '${props.appDetail.api_base_url}/conversations/{conversation_id}/variables/{variable_id}' \
+    --header 'Content-Type: application/json' \
+    --header 'Authorization: Bearer {api_key}' \
+    --data-raw '{
+        "value": "New string value",
+        "user": "abc-123"
+    }'
+    ```
+
+    ```bash {{ title: 'Number Value' }}
+    curl -X PUT '${props.appDetail.api_base_url}/conversations/{conversation_id}/variables/{variable_id}' \
+    --header 'Content-Type: application/json' \
+    --header 'Authorization: Bearer {api_key}' \
+    --data-raw '{
+        "value": 42,
+        "user": "abc-123"
+    }'
+    ```
+
+    ```bash {{ title: 'Object Value' }}
+    curl -X PUT '${props.appDetail.api_base_url}/conversations/{conversation_id}/variables/{variable_id}' \
+    --header 'Content-Type: application/json' \
+    --header 'Authorization: Bearer {api_key}' \
+    --data-raw '{
+        "value": {"product": "Widget", "quantity": 10, "price": 29.99},
+        "user": "abc-123"
+    }'
+    ```
+    </CodeGroup>
+
+    <CodeGroup title="Response">
+    ```json {{ title: 'Response' }}
+    {
+      "id": "variable-uuid-1",
+      "name": "customer_name",
+      "value_type": "string",
+      "value": "Updated Value",
+      "description": "Customer name extracted from the conversation",
+      "created_at": 1650000000000,
+      "updated_at": 1650000001000
+    }
+    ```
+    </CodeGroup>
+  </Col>
+</Row>
+
+---
+
 <Heading
   url='/audio-to-text'
   method='POST'