
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
If the graph of the function \( f(x) \) is symmetric about the points \( (a, b) \) and \( (m, n) \) (where \( a \neq m \)), then for every \( x \) in the domain of the function \( f(x) \), the sequence \( \left\{ y_{k} \right\} \) formed by \( y_{k} = f[x + 2k(a - m)] \) (\( k \in \mathbf{Z} \)) is an arithmetic sequence with a common difference of \( 2(b - n) \).

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