
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 $\{0,1,2,\ldots,n^2\}$ with $a_m \neq 0$. We denote this as \[r+si = (a_ma_{m-1} \ldots a_1a_0)_{-n+i}\].

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

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