
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 \( a, b \in \mathbf{R} \), the circle \( C_{1}: x^{2}+y^{2}-2 x+4 y-b^{2}+5=0 \) and the circle \( C_{2}: x^{2}+y^{2}-2(a-6) x-2 a y +2 a^{2}-12 a+27=0 \) intersect at two distinct points \( A\left(x_{1}, y_{1}\right) \) and \( B\left(x_{2}, y_{2}\right) \), and \( \frac{y_{1}+y_{2}}{x_{1}+x_{2}}+\frac{x_{1}-x_{2}}{y_{1}-y_{2}} = 0 \). Find the value of \( a \).

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