
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 functions $f(x) = -x^3 + x^2 + b$ and $g(x) = a\ln x$,  
(1) If the maximum value of $f(x)$ on the interval $x \in \left[-\frac{1}{2}, 1\right)$ is $\frac{3}{8}$, find the value of the real number $b$.  
(2) If for any $x \in [1,e]$, it is always true that $g(x) \geq -x^2 + (a + 2)x$, find the range of real numbers $a$.  
(3) Under the condition of (1), let $F(x)$ be defined as follows:

$$
F(x) = \begin{cases}
f(x), & x < 1 \\
g(x), & x \geq 1
\end{cases}
$$
For any given positive real number $a$, does there exist two points $P$ and $Q$ on the curve $y = F(x)$ such that triangle $POQ$ is a right-angled triangle with the right angle at the origin $O$, and the midpoint of the hypotenuse lies on the $y$-axis? Please explain your reasoning.

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