
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 the system of inequalities with respect to \( x \):
\[
\begin{cases}
x^3 + 3x^2 - x - 3 > 0, \\
x^2 - 2ax - 1 \leq 0 \quad (a > 0)
\end{cases}
\]
If there is exactly one integer solution, what is the range of values for \( a \)?

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