
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 \( f(x) = x^2 + c \) and \( f(f(x)) = f(x^2 + 1) \),
1. Let \( g(x) = f(f(x)) \). Find the explicit expression for \( g(x) \).
2. Let \( \varphi(x) = g(x) - \lambda f(x) \). Determine the value of the real number \( \lambda \), such that \( \varphi(x) \) is a decreasing function on \( (-\infty, -1] \) and an increasing function on \( [-1, 0) \).

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