
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
If altitude $CE$ is $2\sqrt{2}$ centimeters in $\triangle ABC$, find the area of $\triangle ABC$.

[asy] import olympiad; pair A,B,C,E; A = (0,2*sqrt(2)); B = (2,0);
C = foot(A,B,-B); E = foot(C,A,B); draw(A--B--C--A); draw(C--E,dashed);
label("$45^{\circ}$",A-(0.2,0.3),E);
label("$A$",A,N);label("$B$",B,E);label("$C$",C,W);label("$E$",E,NE);
draw((0,.2)--(.2,.2)--(.2,0)); draw(E + .2*dir(225)--E + sqrt(2)*.2*dir(180)--E+.2*dir(135));
[/asy]

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