
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
Let \( AA' \) and \( BB' \) be two line segments which are perpendicular to \( A'B' \). The lengths of \( AA' \), \( BB' \), and \( A'B' \) are 680, 2000, and 2010 respectively. Find the minimal length of \( AX + XB \) where \( X \) is a point between \( A' \) and \( B' \).

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