
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 $n = 4$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the "base" $-4+i$ using the integers $1,2,\ldots,n^2$ as "digits." That is, the equation \[r+si = a_m(-4+i)^m + a_{m-1}(-4+i)^{m-1} + \cdots + a_1(-4+i) + a_0\] is true for a unique choice of non-negative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from the set $\{0,1,2,\ldots,16\}$, with $a_m \ne 0$. We write \[r+si=(a_ma_{m-1}\ldots a_1a_0)_{-4+i}\] to denote the base $-4+i$ expansion of $r+si$.

There are only finitely many integers $k+0i$ that have three-digit expansions
\[k=(a_2a_1a_0)_{-4+i} \quad (a_2 \neq 0).\] Find the sum of all such $k$.

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