
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 $CD$ is $\sqrt{2}$ centimeters, what is the number of square centimeters in the area of $\Delta ABC$?

[asy] import olympiad; pair A,B,C,D; A = (0,sqrt(2)); B = (1,0);
C = foot(A,B,-B); D = foot(C,A,B); draw(A--B--C--A); draw(C--D,dashed);
label("$45^{\circ}$",A-(0.05,0.5),E);
label("$A$",A,N);label("$B$",B,E);label("$C$",C,W);label("$D$",D,SW);
draw((0,.1)--(.1,.1)--(.1,0)); draw(D + .1*dir(225)--D + sqrt(2)*.1*dir(180)--D+.1*dir(135));
[/asy]

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