
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
In any isosceles triangle $ABC$ with $AB=AC$, the altitude $AD$ bisects the base $BC$ so that $BD=DC$. Given that $AB=AC=17$ and base $BC=16$, determine the area of $\triangle ABC$. Also, if a point $E$ is located 8 units from D along the line extended from D through C such that $DE=8$, find the area of the triangle $AEC$.

[asy]
draw((0,0)--(16,0)--(8,15)--cycle,black+linewidth(1));
draw((8,15)--(8,0),black+linewidth(1)+dashed);
draw((8,0)--(24,0),black+linewidth(1)+dashed); // Extended line to E
draw((8,0)--(8,1)--(9,1)--(9,0)--cycle,black+linewidth(1));
label("$A$",(8,15),N);
label("$B$",(0,0),SW);
label("$C$",(16,0),SE);
label("$D$",(8,0),S);
label("$E$",(24,0),SE);
label("17",(0,0)--(8,15),NW);
label("17",(8,15)--(16,0),NE);
label("16",(8,-1),S);
[/asy]

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