Why is Pakistan involved in a US-Iran peace deal?

Pakistan has played a key role in efforts to halt war between Iran and the United States, with Prime Minister Shehbaz Sharif announcing a deal and “immediate and permanent” end to military operations this week.
Pakistan signalled it will continue to lay the groundwork for further talks, with a signing ceremony scheduled in Switzerland on June 19.
– How did Pakistan take a central role? –
In March, several weeks into the war, Islamabad confirmed it was relaying messages back and forth between Tehran and Washington.
Pakistan’s push to get the warring sides to the negotiating table was met with praise and some surprise.
For those watching closely, however, the development made sens...