
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 \( f(x) = \begin{cases} 
-2x & \text{if } x < 0, \\ 
x^2 - 1 & \text{if } x \geqslant 0 
\end{cases} \), the equation \( f(x) + 2 \sqrt{1 - x^2} + |f(x) - 2 \sqrt{1 - x^2}| - 2ax - 4 = 0 \) has three real roots \( x_1 < x_2 < x_3 \). If \( x_3 - x_2 = 2(x_2 - x_1) \), then the real number \( a \) is _______.

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