
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 $f(x) = \begin{cases} \sqrt{x}+3, & x\geq 0 \\ ax+b, & x<0 \end{cases}$, it satisfies the condition: for any non-zero real number $x_1$, there exists a unique non-zero real number $x_2$ ($x_2 \neq x_1$) such that $f(x_1) = f(x_2)$. When $f(\sqrt{3}a) = f(4b)$ holds, calculate the value of the real number $a+b$.

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