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

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

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