
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 individual plans to distribute two types of products, A and B. According to a survey, when the investment amount is $x$ (where $x \geq 0$) in ten thousand yuan, the profits obtained from distributing products A and B are $f(x)$ and $g(x)$ in ten thousand yuan, respectively. Here, $f(x) = x + 1$; $g(x) = \begin{cases} \frac{10x + 1}{x + 1} & (0 \leq x \leq 3) \\ -x^2 + 9x - 12 & (3 < x \leq 5) \end{cases}$. If the individual is ready to invest 5 ten thousand yuan in these two products, please help him formulate an investment plan to maximize his profit and calculate the maximum profit.

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