
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
The quadratic function \( f(x) = a + bx - x^2 \) satisfies \( f(1+x) = f(1-x) \) for all real numbers \( x \) and \( f(x+m) \) is an increasing function on \( (-\infty, 4] \). Find the range of the real number \( m \).

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