
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
Find the set of values for \( a \) such that the system of equations has a unique solution:
\[
\begin{cases}
|x| + \log_2 (4y + a^2) = 3 \\
\log_{2a^2} (x^2 + 1) + y^2 = 1
\end{cases}
\]
with the constraints \( a > 0 \), \( a \neq 1 \), and \( 4y + a^2 > 0 \).

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