
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 $f(n)$ represents the largest odd factor of a positive integer $n$, for example, the factors of 12 are 1, 2, 3, 4, 6, 12, so $f(12)=3$; the factors of 21 are 1, 3, 7, 21, so $f(21)=21$, find the value of $$\sum\limits_{i=51}^{100}f(i)$$ - $$\sum\limits_{i=1}^{50}f(i)$$.

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