
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
A plane is divided by $n$ lines, which can split the plane into $f(n)$ parts at most.

(1) Find the values of $f(1)$, $f(2)$, $f(3)$, and $f(4)$;

(2) Observe the patterns in $f(2)-f(1)$, $f(3)-f(2)$, and $f(4)-f(3)$;

(3) Determine $f(n)$.

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