
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 \( f(x)=1-|1-2x| \) on the domain \([0,1]\) and \( g(x)=(x-1)^{2} \), and \( F(x) \) is defined as:
\[
F(x) = 
\begin{cases} 
g(x), & \text{if } f(x) \geq g(x); \\ 
f(x), & \text{if } f(x) < g(x).
\end{cases}
\]
Find: 
1. The analytical expression for \( F(x) \).
2. The maximum value of \( F(x) \).
3. The solution to the equation \( F(x) = \frac{1}{3} \).

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