
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
In the diagram below, three squares are inscribed in right triangles. Their areas are  $A$ ,  $M$ , and  $N$ , as indicated in the diagram. If  $M = 5$  and  $N = 12$ , then  $A$  can be expressed as  $a + b\sqrt{c}$ , where  $a$ ,  $b$ , and  $c$  are positive integers and  $c$  is not divisible by the square of any prime. Compute  $a + b + c$ .

[asy]
size(250);
defaultpen (linewidth (0.7) + fontsize (10));
		 pair O = origin, A = (1, 1), B = (4/3, 1/3), C = (2/3, 5/3), P = (3/2, 0), Q = (0,3);
		 draw (P--O--Q--cycle^^(0, 5/3)--C--(2/3,1)^^(0,1)--A--(1,0)^^(1,1/3)--B--(4/3,0));
		 label(" $A$ ", (.5,.5));
		 label(" $M$ ", (7/6, 1/6));
		 label(" $N$ ", (1/3, 4/3));[/asy]

*Proposed by Aaron Lin*

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