
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 \( A = (0, 7) \) and \( B = (0, 14) \). Points \( A' \) and \( B' \) are on the line \( y = x \), and lines \( \overline{AA'} \) and \( \overline{BB'} \) intersect at \( C = (3, 5) \). What is the length of \( \overline{A'B'} \)?

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