
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 $$f(x)= \begin{cases} \sqrt{x}+3, & x\geq 0 \\ ax+b, & x<0 \end{cases}$$, for which it satisfies the condition: for every $x_1 \in \mathbb{R}$, and $x_1 \neq 0$, there exists a unique $x_2 \in \mathbb{R}$ and $x_1 \neq x_2$, such that $f(x_1) = f(x_2)$. Find the sum of the real numbers $a+b$ when $f(2a) = f(3b)$.

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