Retrieving session cookie from C#- silverlight application

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C#

Silverlight

I have been told to forward a cookie from a web app to an authentication service. I can't see the cookie on the file generated when exporting the browser cookies, so when I asked about it, the client told me this is a session cookie. I have searched for information about how to read session cookies, but I find it confusing.

What I have tried:

As far as I understand, ASP.net keeps these cookies in the session dictionary (c# How to get session from asp.net session cookie) But when I explore the class Session, I can't see any dictionary. Besides that, I understand that to read a cookie, you first have to add it, however this cookie is a corporative cookie that I don't manage. Would it be automatically added to my web app session?

Thanks!