
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 expression for $g(x)$;  
(2) Let $\varphi(x)=g(x)-\lambda f(x)$, is there a real number $\lambda$ such that $\varphi(x)$ is a decreasing function in $(-\infty, -1)$ and an increasing function in $(-1, 0)$?

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