
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
Let integers \(a_{1}, a_{2}, \cdots, a_{2019}\) satisfy \(1 = a_{1} \leq a_{2} \leq \cdots \leq a_{2019} = 99\). Define \( f = \left(a_{1}^{2} + a_{2}^{2} + \cdots + a_{2019}^{2}\right) - \left(a_{1} a_{3} + a_{2} a_{4} + \cdots + a_{2017} a_{2019}\right) \).

Find the minimum value of \(f\), denoted as \(f_{0}\), and determine the number of sequences \(\left(a_{1}, a_{2}, \cdots, a_{2019}\right)\) that satisfy \(f = f_{0}\).

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