
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
An equilateral triangle $ABC$ with side length $3$ units has points $D$ and $E$ on sides $AB$ and $BC$, respectively, such that $AD = BE$. When the triangle is folded along the lines $CD$ and $CE$ such that sides $AC$ and $BC$ coincide, they lie along the line $AC$. Given that $AD$ is one unit shorter than side $AB$, express $AD$ in the form $\sqrt{k} - m$ and find the integer value of $k+m$.

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