
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 the integers \( a_{1}, a_{2}, \cdots, a_{2019} \) satisfy:
\[ 1= a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{2019}=99 . \]
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 \( f_{0} \) of \( f \), and determine the number of arrays \( \left(a_{1}, a_{2}, \cdots, a_{2019}\right) \) for which \( f=f_{0} \).

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