
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 \(f(x) = ax^2 + bx + c\) have its graph symmetric about the y-axis. It is given that \(a + b = 1\), and for any point \((x, y)\) on the graph of \(y = f(x)\), the point \(\left(x, y^2 + 1\right)\) lies on the graph of the function \(g(x) = f(f(x))\).

1. Find the analytical expression of \(g(x)\).
2. Let \(F(x) = g(x) - \lambda f(x)\). Determine if 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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