
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 the quadratic function \( f(x) = a x^2 + b x + c \) have its graph symmetric about the y-axis. Given \( a + b = 1 \), and if the point \((x, y)\) is on the graph of \( y = f(x) \), then the point \(\left( x, y^2 + 1 \right) \) is on the graph of the function \( g(x) = f(f(x)) \).

1. Find the expression for \( g(x) \).
2. Let \( F(x) = g(x) - \lambda f(x) \). Determine whether there exists a real number \( \lambda \) such that \( F(x) \) is a decreasing function in \( \left( -\infty, -\frac{\sqrt{2}}{2} \right] \) and an increasing function in \( \left( -\frac{\sqrt{2}}{2}, 0 \right) \).

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