
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
A self-employed person plans to distribute two types of goods, A and B. According to a survey, when the investment amount is $x(x\geqslant 0)$ ten thousand yuan, the profits obtained from distributing goods A and B are $f(x)$ ten thousand yuan and $g(x)$ ten thousand yuan, respectively, where $f(x)=a(x-1)+2(a > 0)$; $g(x)=6\ln (x+b)$, $(b > 0)$. It is known that when the investment amount is zero, the profit is zero.  
$(1)$ Try to find the values of $a$ and $b$;  
$(2)$ If the self-employed person is ready to invest 5 ten thousand yuan in these two types of goods, please help him develop a capital investment plan to maximize his profit, and calculate the maximum value of his income (accurate to $0.1$, reference data: $\ln 3\approx1.10$).

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