
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 now shows 40 lattice points, arranged in an 8 by 5 grid with each point spaced one unit from its neighbors. A new segment $\overline{AB}$ stretches from point $A$ at (0,4) to point $B$ at (7,0). Another segment $\overline{CD}$ extends from point $C$ at (5,0) to point $D$ at (6,4). The two segments meet at point $E$. Determine the length of segment $AE$.
[asy]
unitsize(1cm);
for (int i=0; i<8; ++i) {
for (int j=0; j<5; ++j) {
dot((i,j));
}}
label("$A$", (0,4), W);
label("$B$", (7,0), E);
label("$C$", (5,0), S);
label("$D$", (6,4), N);
draw((0,4)--(7,0),linewidth(0.7));
draw((5,0)--(6,4),linewidth(0.7));
[/asy]

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