
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
Rectangle $ABCD$ has side lengths $2$ units (AB, CD) and $1$ unit (BC, AD). Points $E$ and $F$ are on sides $AB$ and $CD$, respectively, such that $AE = DF = x$. When the rectangle is folded along lines $DE$ and $BF$, points $A$ and $D$ coincide at point $G$ on $BD$, a diagonal. Find $x$ such that $AD$ coincides with $CD$ on $BD$ when folded, and express $x$ in the form $\sqrt{k}-m$ units. What is $k+m$?

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