The shapes are reminiscent of tangram puzzles, so we turn to the Tangrams interactive manual and try to recreate these shapes. We quickly notice that the digits tell us which pieces to use, and the arrows indicate how to rotate them (the arrow points where the white dot needs to go).

We end up with the arrangements shown below.

We can then connect the red circles with the uncircled numbers by following the lines in the diagrams (while observing the rule about the diodes which restrict the direction in which current can flow). These connections are shown in different colors below.

The circled numbers run consecutively from 1 to 12, so we can order them in this way. Furthermore, the flavor text suggests to add up the digits that are reachable.

#

Reachable

Sum

Letter

1

9, 9

18

R

2

1

1

A

3

9, 8, 5

22

V

4

5

5

E

5

2, 5, 3, 4

14

N

6

4, 6, 8, 1

19

S

7

1, 1

2

B

8

7, 8, 6

21

U

9

9, 9

18

R

10

1, 6

7

G

11

5

5

E

12

9, 9

18

R

Converting the sums into letters using A1Z26 reveals the answer.