
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
Find all the ways of placing the integers  $1,2,3,\cdots,16$  in the boxes below, such that each integer appears in exactly one box, and the sum of every pair of neighboring integers is a perfect square. 
[asy]
import graph;
real r=10;
size(r*cm);
picture square1;
draw(square1, (0,0)--(0,1)--(1,1)--(1,0)--cycle);
add(scale(r/31*cm)*square1,(0,0));
picture square2;
draw(square2, (-1,0.5)--(0,0.5)--(0,1)--(1,1)--(1,0)--(0,0)--(0,0.5));
for(int i=1; i<16; ++i)
{
add(scale(r/31*cm)*square2,(i,0));
}[/asy]

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