
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
Given circles $C_{1}{:}\ x^{2}{+}y^{2}{-}x{+}y{-}2{=}0$ and $C_{2}{:}\ x^{2}{+}y^{2}{=}5$ intersect at points A and B,

1. Find the equation of the line containing chord AB, where circles $C_{1}$ and $C_{2}$ intersect.
2. Find the length of the common chord AB shared by circles $C_{1}$ and $C_{2}$.
3. Find the equation of the perpendicular bisector of line segment AB.

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