
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 numbers 1 through 10 can be arranged along the vertices and sides of a pentagon so that the sum of the three numbers along each side is the same. The diagram below shows an arrangement with sum 16. Find, with proof, the smallest possible value for a sum and give an example of an arrangement with that sum.

[asy]
int i;
pair[] A={dir(18+72*0), dir(18+72*1), dir(18+72*2),dir(18+72*3), dir(18+72*4), dir(18+72*0)};
pair O=origin;
int[] v = {7,1,10,4,3};
int[] s = {8, 5, 2, 9, 6};
for(i=0; i<5; i=i+1) {
label(string(v[i]), A[i], dir(O--A[i]));
label(string(s[i]), A[i]--A[i+1], dir(-90)*dir(A[i]--A[i+1]));
}
draw(rotate(0)*polygon(5));[/asy]

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