
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 a triangle \(ABC\), the angle bisector from \(A\), the median from \(B\), and the altitude from \(C\) are concurrent and furthermore the angle bisector from \(A\) and the median from \(B\) are perpendicular. If side \(AB\) measures one unit, find the lengths of the other two sides.

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