How far can the Abbasids go in the 1200s?

After centuries of decline and subjugation at the hands of foreign powers like the Buyids and the Seljuk Turks, the Abbasid Caliphate took advantage of the latter's decline in the 12th century to reassert its independence, forming a state in what is now Iraq. This resurgence came to an end, however, after the arrival of the Mongols, who captured Baghdad and massacred its inhabitants after a short siege in 1258.

Assuming the Mongols aren't a factor (maybe they never manage to break out of eastern Asia or focus their expansionism elsewhere) or that the Abbasids somehow manage to defeat them, how could the Caliphate evolve in the upcoming decades? Could they expand out of Mesopotamia?
 
According to the wikipedia article on the siege of Baghdad, the Abbasids made no preparations against the Mongols during the time that preceded their arrival. Was there anything they could've done to save their capital? Did they have any foreign allies, like the Egyptian Mamluks who would eventually shelter the dynasty in Cairo?
 
According to the wikipedia article on the siege of Baghdad, the Abbasids made no preparations against the Mongols during the time that preceded their arrival. Was there anything they could've done to save their capital? Did they have any foreign allies, like the Egyptian Mamluks who would eventually shelter the dynasty in Cairo?
IIRC the abbasid caliph thought that god would send a plague upon the mongols and prevent them from fighting him.
 
Even if we count off the Mongols there was another closer to home threat, the Khwarezmians. The Khwarezmians were so antagonistic to Abbasids that the latter congratulated Genghis khan after he crushed them. If we butterfly mongols and have the mediocre Alauddin rule over Khurasan, then we can hope to have a stable Abbasid realm roughly comprising areas of modern iraq. At this point there is no chance whatsoever at controlling the holy cities, which would give the Abbasids prestige. Of course, if Jalaluddin ever comes to power, Abbasids are good as gone.
 
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