
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
The diagram shows two equilateral triangles with side length  $4$  mounted on two adjacent sides of a square also with side length  $4$ . The distance between the two vertices marked  $A$  and  $B$  can be written as  $\sqrt{m}+\sqrt{n}$  for two positive integers  $m$  and  $n$ . Find  $m + n$ .

[asy]
size(120);
defaultpen(linewidth(0.7)+fontsize(11pt));
draw(unitsquare);
draw((0,1)--(1/2,1+sqrt(3)/2)--(1,1)--(1+sqrt(3)/2,1/2)--(1,0));
label(" $A$ ",(1/2,1+sqrt(3)/2),N);
label(" $B$ ",(1+sqrt(3)/2,1/2),E);
[/asy]

Present the answer in LaTex format: \boxed{Your answer}
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