
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 \( f(x) = ax^2 + bx + c \) (where \( a, b, c \in \mathbb{R} \) and \( a \neq 0 \)) satisfy the following conditions:
1. For all \( x \in \mathbb{R} \), \( f(x-4) = f(2-x) \);
2. For \( x \in (0,2) \), \( f(x) \leq \left( \frac{x+1}{2} \right)^2 \);
3. The minimum value of \( f(x) \) over \( \mathbb{R} \) is 0. 

Find the maximum value of \( m \) (with \( m > 1 \)) such that there exists \( t \in \mathbb{R} \) where for all \( x \in [1, m] \), \( f(x+t) \leq x \).

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