docs: document the Ask AI chat widget's bottom prompt trigger

fern/products/docs/pages/ask-fern/ask-ai-chat.mdx

@@ -1,6 +1,6 @@
 ---
 title: Ask AI chat panel
-description: Embed Fern's Ask AI chat panel in any React application, with a floating, inline, or custom trigger.
+description: Embed Fern's Ask AI chat panel in any React application, with a floating launcher, a bottom prompt, an inline button, or a custom trigger.
 ---
 
 `AskAiChat` renders a sidebar panel with AI chat (no keyword search). It's one of the two components in the [`@fern-api/search-widget` package](/learn/docs/ai-features/ask-fern/search-widget/overview), which covers the prerequisites and the Content Security Policy the widget needs.
@@ -51,7 +51,7 @@ description: Embed Fern's Ask AI chat panel in any React application, with a flo
     ```
   </Step>
   <Step title="Match the chat to your product">
-    `theme` controls light and dark mode, and `accentColor` sets the accent used for the sparkles, the send button, and links in answers, in both modes. `trigger` chooses between the corner launcher (`floating`), an unpainted button you place and style yourself (`inline`), and no button at all (`none`).
+    `theme` controls light and dark mode, and `accentColor` sets the accent used for the sparkles, the send button, and links in answers, in both modes. `trigger` chooses between the corner launcher (`floating`), a prompt input centered at the bottom of the viewport (`prompt`), an unpainted button you place and style yourself (`inline`), and no button at all (`none`).
 
     ```tsx
     <AskAiChat domain="https://docs.example.com" theme="inherit" accentColor="#6b4eff" />
@@ -94,6 +94,7 @@ description: Embed Fern's Ask AI chat panel in any React application, with a flo
     *  `AskAiChat.Root` holds the state and takes the configuration props: `domain`, `lang`, `searchLocale`, `placement`, `theme`, `accentColor`, `open`, `defaultOpen`, and `onOpenChange`. Its `children` is the subtree to wrap, the menu and the panel.
     *  `AskAiChat.Trigger` is a real `<button>`, so `asChild` works and the menu keeps its keyboard behavior. Styling props (`className`, `style`, `icon`, and any button attributes) go here. `variant="floating"` renders the corner launcher instead of an unpainted button.
     *  `AskAiChat.Panel` is the chat. Render exactly one per root, outside any menu.
+    *  `AskAiChat.Prompt` is the bottom prompt input, an alternative to a trigger. It takes `placeholder`, `icon`, `className`, and any form attributes.
 
     The same shape covers a popover, a command palette, or a nav bar dropdown, and a single root supports multiple triggers, at most one of them `variant="floating"`.
 
@@ -160,8 +161,8 @@ All standard HTML button attributes are also supported and forwarded to the trig
   Accent color for the sparkles, send button, and links in answers. Accepts any CSS color (for example, `#6b4eff`), and derives the light and dark variants from it. Defaults to a neutral gray.
 </ParamField>
 
-<ParamField path="trigger" type="'floating' | 'inline' | 'none'" default="floating">
-  `floating` renders a styled launcher pinned to a corner of the viewport. `inline` renders an unpainted button to place and style yourself, such as in a nav bar. Both carry the sparkle icon and an `Ask AI` label by default. `none` renders no trigger at all, for [driving the open state from your own UI](#custom-triggers).
+<ParamField path="trigger" type="'floating' | 'prompt' | 'inline' | 'none'" default="floating">
+  `floating` renders a styled launcher pinned to a corner of the viewport. `inline` renders an unpainted button to place and style yourself, such as in a nav bar. Both carry the sparkle icon and an `Ask AI` label by default. `prompt` renders a single-line input pinned to the bottom center of the viewport instead of a button, and sending a question opens the chat directly above it with that question already in flight. It ignores `placement`, sizes the input with `--fern-ask-ai-prompt-width` and `--fern-ask-ai-prompt-width-focus`, and the chat it opens with `--fern-ask-ai-prompt-card-width`. Below 768px the chat is the same bottom drawer the corner launcher opens. `none` renders no trigger at all, for [driving the open state from your own UI](#custom-triggers).
 </ParamField>
 
 <ParamField path="open" type="boolean">
@@ -177,7 +178,7 @@ All standard HTML button attributes are also supported and forwarded to the trig
 </ParamField>
 
 <ParamField path="placement" type="'bottom-right' | 'bottom-left' | 'top-right' | 'top-left'" default="bottom-right">
-  The corner the launcher and card anchor to.
+  The corner the launcher and card anchor to. Ignored when `trigger` is `prompt`.
 </ParamField>
 
 <ParamField path="searchLocale" type="string">