
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 that the graph \( C^{\prime} \) of a certain function is symmetric with respect to the line \( y = x \) to the graph \( C: y = \frac{a x + a^{2} + 1}{x + a - 1} \), and that the graph \( C^{\prime} \) is symmetric with respect to the point \( (3,-2) \), find the value of the real number \( a \).

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