
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
Square $ABCD$ has a right triangle $EFG$ inside it, where $\overline{EF}$ is the hypotenuse, $\overline{EG}$ is vertical and $\overline{FG}$ is horizontal. If $AE = 1$, $EC = BF = 2$, compute $DG$.

[asy]
unitsize(0.8 cm);

pair A, B, C, D, E, F, G;

A = (0,0);
B = (4,0);
C = (4,4);
D = (0,4);
E = (0,2);
F = (2,0);
G = (2,2);

draw(A--B--C--D--cycle);
draw(E--F--G--cycle);
draw(rightanglemark(E,G,F,10));

label("$A$", A, W);
label("$C$", C, E);
label("$B$", B, S);
label("$D$", D, W);
label("$E$", E, W);
label("$F$", F, S);
label("$G$", G, NE);
label("$2$", (B + F)/2, S);
label("$2$", (E + C)/2, E);
label("$1$", (A + E)/2, W);
[/asy]

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