My first post here so please be kind 

I work for a state hospital and my salary is a combination of the private practice plan and state salary.
Both the state and the plan contribute a percentage of each respective salary to separate 401a accounts with different investment firms. I do not make any contributions to either 401a.
I have access to a 403b plan offered by the state and another one offered by the private practice plan.
I also have access to a government 457b plan to which I have never contributed.
I read that there is a maximum total contribution for 2017 of $54,000/yr for total employer/employee plans. I also read that the total is $54,000/yr for just the 401a plan. It doesn't seem to mention 457 plans.
So my understanding is that I can contribute the following:
24k/yr to each 403b plan
Since I am within 3 years of "normal retirement age" as defined by my employer, I think I can contribute 18k plus 18k catchup to the 457b plan.
Both the state and the practice plan can contribute to their separate 401a.
Questions:
Is the $54k limit an aggregate of both 401a plans or is is 54k for each plan?
Are the 401a and 403b amounts combined for the purpose of the 54k limit?
Is the 457b separate and distinct?
Thanks for your help.
Kara
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