
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 quadratic function $f(x) = -x^2 + bx + c$ defined on $\mathbb{R}$, and $f(x) = f(1-x)$. For the sequence $\{a_n\}$, if $a_1 = 0$ and $a_{n+1} = f(a_n)$ ($n \in \mathbb{N}^*$),  
(1) Find the necessary and sufficient condition for the sequence $\{a_n\}$ to be a monotonically decreasing sequence;  
(2) Find the range of values for $c$ such that the sequence $\{a_n\}$ is a monotonically increasing sequence.

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