
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 a function $f(x) = x^2 + 2ax + b$ whose graph passes through the point $(1,3)$, and $f(-1 + x) = f(-1 - x)$ holds true for all real numbers. The graph of $y = g(x)$ is symmetric to the graph of $y = f(x)$ with respect to the origin.
(1) Find the explicit formulas for $f(x)$ and $g(x)$;
(2) If $F(x) = g(x) - \lambda f(x)$ is increasing on the interval $(-1,1)$, determine the range of the real number $\lambda$.

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