
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
Determine the sum of all integers \(k\) that can be represented as three-digit expansions in the base \(-4+i\) like so:
\[k = (a_2a_1a_0)_{-4+i}\quad \text{(where } a_2 \neq 0)\]
given that \(k\) is a real number and each \(a_i\) is chosen from the set \(\{0, 1, \ldots, 16\}\).

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