
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 $f(x) = axe^x$ has its tangent at point $(0, 0)$ parallel to the line $y = x - 2$.

(Ⅰ) Find the value of the real number $a$;  
(Ⅱ) Let $g(x) = f(x) - b\left( \frac{x^2}{2} + x \right)$  
(i) If the function $g(x) \geq 0$ holds true on $[0, +\infty)$, find the maximum value of the real number $b$;  
(ii) When $b \leq 0$, determine the number of zeros of the function $g(x)$ and provide a proof.

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