
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 that $t \neq 0$, point $P(t, 0)$ is a common point on the graphs of functions $f(x) = x^3 + ax$ and $g(x) = bx^2 + c$. The graphs of these functions have the same tangent line at point $P$. (I) Express $a$, $b$, and $c$ in terms of $t$; (II) If the function $y = f(x) - g(x)$ is monotonically decreasing on the interval $(-1, 3)$, determine the range of possible values for $t$.

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