Eldritch Knight / Bladesinger multiclass

I have heard that this is a powerful multiclass combo, and we are about to start a campaign where I am starting out as a lvl 5 EK. The DM and I have backstory ties and flavor reasons for me to be able to multiclass at some point into Wizard. My question is:

I know I should get to lvl 7 in EK first, but after that how would you approach the multiclass? I was thinking at least next 3 lvls (8-10) in blade singer, or maybe go all the way to 13 (7 EK, 6 BS), then finish to 20 as fighter. This is all very premature, I’m just pondering my options here and have been reading the new rule book — curious to see what y’all think! No wrong answers, I appreciate any ideas or feedback.

Edit:

Thank you everyone who responded! I have a lot of ideas to ruminate on. I’m rethinking just monoclassing and finding other ways to incorporate the story elements and get more magic (feats and so on). DM is being very collaborative and is finding ways to weave my backstory into the campaign.