
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 integers \( a_1, a_2, \cdots, a_{2011} \) satisfying \( 1 = a_1 \leq a_2 \leq \cdots \leq a_{2019} = 99 \), let 
\[ f = \left( a_1^2 + a_2^2 + \cdots + a_{2019}^2 \right) - \left( a_1 a_3 + a_2 a_4 + a_3 a_5 + \cdots + a_{2017} a_{2019} \right). \]
Find the minimum value \( f_0 \) of \( f \). Also, 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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