
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 rectangle $ABCD$, $AB=7$ and $BC =4$. Points $F$ and $G$ are on $\overline{CD}$ so that $DF = 2$ and $GC=1$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$. [asy]
pair A,B,C,D,I,F,G;
A=(0,0);
B=(7,0);
C=(7,4);
D=(0,4);
F=(2,4);
G=(6,4);
I=(2.2,6.5);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(A--B--I--cycle,linewidth(0.7));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$F$",F,SE);
label("$G$",G,SW);
label("$E$",I,N);
label("2",(1,4),N);
label("1",(6.5,4),N);
label("4",(0,2),W);
label("4",(7,2),E);
label("7",(3.5,0),S);
[/asy]

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