
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 side length $1$ unit. Points $E$ and $F$ are on sides $AB$ and $CB$ respectively, with $AE = x$ and $CF = 1/2$. When the square is folded along the lines $DE$ and $DF$, sides $AD$ and $CD$ coincide and lie on diagonal $BD$. Find $AE$, which can be expressed as $\sqrt{k}-m$ units and determine the integer value of $k+m$.

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