
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 the circles $C_{1}: x^{2}+y^{2}-2x+4y-4=0$, $C_{2}: x^{2}+y^{2}+2x+2y-2=0$, and $C_{3}: x^{2}+y^{2}-2x-2y-\frac{14}{5}=0$, find the length of the chord that lies on the line of the common chord of circles $C_{1}$ and $C_{2}$ and is intercepted by circle $C_{3}$.

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