
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 a positive integer \( n \), every complex number of the form \( r+si \), where \( r \) and \( s \) are integers, can be uniquely expressed in the "base" \( -n+i \) using the integers \( 1,2,\ldots,n^2 \) as "digits." Specifically, the equation
\[
r+si = a_m(-n+i)^m + a_{m-1}(-n+i)^{m-1} + \cdots + a_1(-n+i) + a_0
\]
holds for a unique choice of non-negative integer \( m \) and digits \( a_0, a_1, \ldots, a_m \) chosen from the set \( \{0,1,2,\ldots,n^2\} \), with \( a_m \neq 0 \). We write 
\[
r+si = (a_ma_{m-1}\ldots a_1a_0)_{-n+i}
\]
to denote the base \( -n+i \) expansion of \( r+si \).

Find the sum of all integers \( k \) that have four-digit expansions
\[
k = (a_3a_2a_1a_0)_{-4+i} \qquad (a_3 \neq 0).
\]

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