
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
If a coordinate origin $O$ is taken as an acute vertex to form an isosceles right triangle inscribed in a parabola $y=a x^{2}$ (where $a>0$), and the right-angle vertex $C$ is in the first quadrant, the ratio of the vertical and horizontal coordinates of point $C$, $t\left(=\frac{y_{c}}{x_{c}}\right)$, must satisfy a cubic equation expressed in terms of $t$. What is the cubic equation?

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