
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
Let function \( f(x) = 1 - |1 - 2x| \) and \( g(x) = x^2 - 2x + 1 \) for \( x \in [0,1] \), and define

\[ 
F(x) = 
\begin{cases} 
f(x) & \text{if } f(x) \geq g(x), \\
g(x) & \text{if } f(x) < g(x). 
\end{cases}
 \]

Determine the number of real roots of the equation \( F(x) \cdot 2^x = 1 \).

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