--- Part Two ---

You arrive at the vault only to discover that there is not one vault, but four - each with its own
entrance.

On your map, find the area in the middle that looks like this:

...
.@.
...

Update your map to instead use the correct data:

@#@
###
@#@

This change will split your map into four separate sections, each with its own entrance:

#######       #######
#a.#Cd#       #a.#Cd#
##...##       ##@#@##
##.@.##  -->  #######
##...##       ##@#@##
#cB#Ab#       #cB#Ab#
#######       #######

Because some of the keys are for doors in other vaults, it would take much too long to collect all
of the keys by yourself.  Instead, you deploy four remote-controlled robots. Each starts at one of
the entrances (@).

Your goal is still to collect all of the keys in the fewest steps, but now, each robot has its own
position and can move independently.  You can only remotely control a single robot at a time.
Collecting a key instantly unlocks any corresponding doors, regardless of the vault in which the key
or door is found.

For example, in the map above, the top-left robot first collects key a, unlocking door A in the
bottom-right vault:

#######
#@.#Cd#
##.#@##
#######
##@#@##
#cB#.b#
#######

Then, the bottom-right robot collects key b, unlocking door B in the bottom-left vault:

#######
#@.#Cd#
##.#@##
#######
##@#.##
#c.#.@#
#######

Then, the bottom-left robot collects key c:

#######
#@.#.d#
##.#@##
#######
##.#.##
#@.#.@#
#######

Finally, the top-right robot collects key d:

#######
#@.#.@#
##.#.##
#######
##.#.##
#@.#.@#
#######

In this example, it only took 8 steps to collect all of the keys.

Sometimes, multiple robots might have keys available, or a robot might have to wait for multiple
keys to be collected:

###############
#d.ABC.#.....a#
######@#@######
###############
######@#@######
#b.....#.....c#
###############

First, the top-right, bottom-left, and bottom-right robots take turns collecting keys a, b, and c, a
total of 6 + 6 + 6 = 18 steps. Then, the top-left robot can access key d, spending another 6 steps;
collecting all of the keys here takes a minimum of 24 steps.

Here's a more complex example:

#############
#DcBa.#.GhKl#
#.###@#@#I###
#e#d#####j#k#
###C#@#@###J#
#fEbA.#.FgHi#
#############


 - Top-left robot collects key a.

 - Bottom-left robot collects key b.

 - Top-left robot collects key c.

 - Bottom-left robot collects key d.

 - Top-left robot collects key e.

 - Bottom-left robot collects key f.

 - Bottom-right robot collects key g.

 - Top-right robot collects key h.

 - Bottom-right robot collects key i.

 - Top-right robot collects key j.

 - Bottom-right robot collects key k.

 - Top-right robot collects key l.


In the above example, the fewest steps to collect all of the keys is 32.

Here's an example with more choices:

#############
#g#f.D#..h#l#
#F###e#E###.#
#dCba@#@BcIJ#
#############
#nK.L@#@G...#
#M###N#H###.#
#o#m..#i#jk.#
#############

One solution with the fewest steps is:


 - Top-left robot collects key e.

 - Top-right robot collects key h.

 - Bottom-right robot collects key i.

 - Top-left robot collects key a.

 - Top-left robot collects key b.

 - Top-right robot collects key c.

 - Top-left robot collects key d.

 - Top-left robot collects key f.

 - Top-left robot collects key g.

 - Bottom-right robot collects key k.

 - Bottom-right robot collects key j.

 - Top-right robot collects key l.

 - Bottom-left robot collects key n.

 - Bottom-left robot collects key m.

 - Bottom-left robot collects key o.


This example requires at least 72 steps to collect all keys.

After updating your map and using the remote-controlled robots, what is the fewest steps necessary
to collect all of the keys?