
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 \( a, b > 0 \) satisfying the equation \( \sqrt{|x|} + \sqrt{|x + a|} = b \) having exactly three distinct real solutions \( x_{1}, x_{2}, x_{3} \), with \( x_{1} < x_{2} < x_{3} = b \), find the value of \( a + b \).

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