
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
Let \( a \) be a positive number. Consider the system of equations:

\[
\begin{cases}
y = ax^2 \\
y^2 + 3 = x^2 + 4y
\end{cases}
\]

Determine the number of different real solutions to the system.

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