
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 coefficients of the following cubic polynomial

\[ 
f(x) \equiv Ax^{3}+Bx^{2}+Cx+D
\]

given that when \( x = 1 \), \( f(1) = 0 \), and

\[ 
\begin{aligned}
x = \frac{2}{3}, \quad f\left(\frac{2}{3}\right) = -4, \\
x = \frac{4}{5}, \quad f\left(\frac{4}{5}\right) = -\frac{16}{5}
\end{aligned}
\]

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