
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: On the graph of the function, $f(x) = mx^3 - x$ has a tangent at point $N(1, n)$ with an inclination angle of $\frac{\pi}{4}$.

(I) Find the values of $m$ and $n$;  
(II) Is there a smallest positive integer $k$ such that the inequality $f(x) \leq k - 1993$ holds for all $x \in [-1, 3]$? If it exists, find the smallest positive integer $k$; if not, explain why.

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