fix: end a spent tool budget in an answer, not a notice about the loop

Asked for the fish in every Valheim biome, the turn researched for five
rounds and then displayed "Stopped after 5 rounds of tool calls without a
final answer." Three things compounded:

- MAX_TOOL_ITERATIONS of 5 is too few for a question that needs several
  searches and a few pages. It is a safety net against looping, not a
  budget anyone chose.
- The loop discarded everything. `content` was scoped to the iteration, so
  every preamble the model wrote was thrown away and the turn ended with an
  app-generated message about its own control flow.
- The model was never told it was running out, so it kept searching until
  the hard stop instead of answering with what it had.

The budget is now spent on a closing round with the tools taken away and an
instruction to answer from what was gathered, naming anything it could not
confirm. So the worst case is a partial answer that says what is missing.
If the closing round produces nothing, the last preamble is used, and only
then the notice. The cap is 8 because the extra round is not more searching,
it is the answer.

Verified: 349 tests, tsc, eslint, prettier, and vici build.

README.md

@@ -16,7 +16,7 @@ Hit a global shortcut, type a question, get a streamed answer without leaving wh
 - **Persistent conversations.** Every chat is saved locally. The sidebar lists them by recency, and reopening the command picks up where you left off, unless it has been idle longer than you allow.
 - **Readable code.** Fenced blocks render in a dark panel with the language named in a header, syntax colouring, and a Copy button.
 - **Real markdown.** Tables, bulleted and numbered lists, headings, blockquotes, bold, italics and links all render natively, and the default prompt asks for the shape that fits the answer rather than flattening everything into prose.
-- **Tools.** Lam decides when to call them: current time, location context, web search, and page fetching.
+- **Tools.** Lam decides when to call them: current time, location context, web search, and page fetching. A turn that uses up its tool budget still ends in an answer, written from what it gathered, rather than a notice that it stopped.
 - **Screenshots, two ways.** Snip a region into the conversation you are already in from the action panel, or run **Snip to Lam** to start a new chat about it. The launcher window gets out of the way first, so it never ends up in the shot, and the capture appears in the transcript as a thumbnail with Open Screenshot for the full image. Neither is offered unless the current model can actually see images.
 - **Chat only.** Lam answers questions. It does not touch your files or run commands. The one exception is capture, and it only ever happens because you pressed the key for it.
 
@@ -158,7 +158,9 @@ This extension does not use GPS. That would mean CoreLocation on macOS, which ha
 
 **The provider has to be one that answers an unauthenticated caller.** The first choice was ipapi.co, which no longer does: a request with no User-Agent now returns `HTTP 429` with `"Please sign up for a paid plan"`, and adding a browser User-Agent gets an `HTTP 403` Cloudflare bot challenge instead. ipwho.is answers without a key and signals failures with `success: false` on an `HTTP 200`, so the body is checked as well as the status. `GEO_PROVIDER` is a single constant, and swapping it means changing the parser beside it.
 
-Web search results and fetched pages can be large. Lam truncates them before they reach the model, and gives up after 5 rounds of tool calls rather than looping.
+Web search results and fetched pages can be large. Lam truncates them before they reach the model.
+
+**A turn that runs out of tool calls still ends in an answer.** The loop stops after 8 rounds, and the first version of it stopped by throwing away every preamble the model had written and returning a notice about the loop: asked for the fish in every Valheim biome, it researched for five rounds and then said nothing useful at all. The budget is now spent on a *closing round* with the tools taken away and an instruction to answer from what was gathered and to name what it could not confirm. So the worst case is a partial answer that says what is missing, rather than an app-generated message about its own control flow. If the closing round produces nothing either, the last preamble is used, and only then a notice. The closing round is why the cap is 8 rather than 5: the extra round is not more searching, it is the answer.
 
 ## Architecture