
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 that the function $f(x)$ is monotonic on $(-1, +\infty)$, and the graph of the function $y = f(x - 2)$ is symmetrical about the line $x = 1$, if the sequence $\{a_n\}$ is an arithmetic sequence with a nonzero common difference and $f(a_{50}) = f(a_{51})$, determine the sum of the first 100 terms of $\{a_n\}$.

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