
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 function \( f(x)=\left\{\begin{array}{ll}
-2x, & x<0 ; \\
x^{2}-1, & x \geqslant 0
\end{array}\right. \), solve the equation:

\[ f(x) + 2\sqrt{1-x^2} + \left| f(x) - 2\sqrt{1-x^2} \right| - 2ax - 4 = 0 \]

There are three real roots \( x_1 < x_2 < x_3 \). If \( x_3 - x_2 = 2(x_2 - x_1) \), then the real number \( a = \) ____.

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