
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
For each positive digit \( D \) and positive integer \( k \), we use the symbol \( D_{(k)} \) to represent the positive integer having exactly \( k \) digits, each of which is equal to \( D \). For example, \( 2_{(1)} = 2 \) and \( 3_{(4)} = 3333 \). Find the sum of the digits of the number of quadruples \( (P, Q, R, k) \) with \( P, Q \) and \( R \) positive digits, \( k \) a positive integer with \( k \leq 2018 \), and \( P_{(2k)} - Q_{(k)} = \left(R_{(k)}\right)^2 \).

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