
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
Suppose the quadratic function \( f(x)=a x^{2}+b x+c \) (where \( a, b, c \in \mathbf{R} \) and \( a \neq 0 \)) satisfies the following conditions:
(1) For \( x \in \mathbf{R} \), \( f(x-4)=f(2-x) \), and \( f(x) \geqslant x \);
(2) For \( x \in(0,2) \), \( f(x) \leqslant \left(\frac{x+1}{2}\right)^{2} \);
(3) The minimum value of \( f(x) \) on \( \mathbf{R} \) is 0.

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

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