
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 value of 
\[
\sum_{k = 1}^{50} (-1)^k \cdot \frac{k^3 + k^2 + 1}{k!}
\]
and express it in the form \(\frac{a}{b!} - c\), where \( a, b, \) and \( c \) are positive integers. Find \( a + b + c \).

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