
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
A 5x5 geoboard is shown with points evenly spaced vertically and horizontally. Segment $DE$ is drawn as shown. Point $F$ is to be chosen from the remaining $23$ points on the grid. How many of these $23$ points will result in triangle $DEF$ being isosceles if $DE$ is four units long? (Note: $D$ and $E$ are positioned at one end of a row, for instance $(1,3)$ and $(5,3)$, respectively).
[asy]
draw((0,0)--(0,6)--(6,6)--(6,0)--cycle,linewidth(1));
for(int i=1; i<6; ++i)
{
  for(int j=1; j<6; ++j)
  {
    dot((i,j));
  }
}
draw((1,3)--(5,3),linewidth(1));
label("D", (1,3), SW);
label("E", (5,3), SE);
[/asy]

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