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I would do this with TWO virtual servers:

Port 80 (HTTP) VIP with nothing but an HTTP profile and the built-in HTTP-to-HTTPS iRule. This will take all HTTP:// port requests and automatically redirect them to the HTTPS virtual server. No pool or other profiles necessary on this VIP.

Port 443 (HTTPS) VIP with your application pool, client SSL profile, any other profiles and settings, and the following iRule:

when HTTP_REQUEST {
     if { [HTTP::uri] equals "/" } {
          HTTP::respond 302 Location "/qqest"
     } elseif { [string tolower [HTTP::uri]] starts_with "/qqest" } {
          pool PEOtime
     } else {
          pool PEOsummit
     }
}