Make sure stdout is clean during all request processing

RuboCop is still too chatty and breaks the protocol by printing to
stdout despite our best attempts. Today I was having problems with the
formatting on Tapioca and realized it was because of the:
```
An error occurred while Layout/BlockEndNewline cop was inspecting /Users/ufuk/src/github.com/Shopify/tapioca/lib/tapioca/static/rbs_converter.rb:166:8.
To see the complete backtrace run rubocop -d.
```
problem (which is in per-1.34 versions of RuboCop). I am not exactly
sure why this was causing a problem, since that specific message should
be going to stderr, but the LSP communication was still broken because
of it.

Regardless, I realized that we never want any request to print to
stdout, willingly or by mistake, so I decided to resolve that at the
handler layer.

lib/ruby_lsp/queue.rb

@@ -91,7 +91,9 @@ def execute(request)
       error = T.let(nil, T.nilable(StandardError))
 
       request_time = Benchmark.realtime do
-        response = T.must(@handlers[request[:method]]).action.call(request)
+        response = capture_stdout do
+          T.must(@handlers[request[:method]]).action.call(request)
+        end
       rescue StandardError => e
         error = e
       end
@@ -131,6 +133,18 @@ def finalize_request(result, request)
 
     private
 
+    sig do
+      type_parameters(:T).params(block: T.proc.returns(T.type_parameter(:T))).returns(T.type_parameter(:T))
+    end
+    def capture_stdout(&block)
+      orig_stdout = $stdout
+      $stdout = StringIO.new
+
+      block.call
+    ensure
+      $stdout = orig_stdout
+    end
+
     sig { returns(Thread) }
     def new_worker
       Thread.new do