
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, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Determine the length of segment $AE$.

[asy]
unitsize(0.8cm);
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("$D$",(3,0),S);
label("$E$",(4.2,1.7),S);
dot((4.2,1.7));
label("$C$",(5,3),N);
draw((0,4)--(7,0),linewidth(0.7));
draw((3,0)--(5,3),linewidth(0.7));
[/asy]

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