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

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