
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 \( x \neq \pm 1 \) and \( x \neq -3 \). If \( a \) is the real root of the equation \(\frac{1}{x-1} + \frac{1}{x+3} = \frac{2}{x^2-1}\), find the value of \( a \).

If \( b > 1 \), \( f(b) = \frac{-a}{\log_2 b} \) and \( g(b) = 1 + \frac{1}{\log_3 b} \). If \( b \) satisfies the equation \(|f(b) - g(b)| + f(b) + g(b) = 3\), find the value of \( b \).

Given that \( x_0 \) satisfies the equation \( x^2 - 5x + (b-8) = 0 \). If \( c = \frac{x_0^2}{x_0^4 + x_0^2 + 1} \), find the value of \( c \).

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