
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 \(a, b > 0\), satisfy the equation for \(x\):
\[ 
\sqrt{|x|} + \sqrt{|x + a|} = b
\]

There are exactly three distinct real solutions \(x_1, x_2, x_3\), where \(x_1 < x_2 < x_3 = b\). Find the value of \(a + b\).

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