
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" $-n+i$. Specifically, consider the equation \[r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0\] 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 $a_m\ne 0$. 

Find the sum of all integers $k+0i$ that have a four-digit expansion in the base $-4+i$, where $k=(a_3a_2a_1a_0)_{-4+i}$ and $a_3 \neq 0$.

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