
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 function
$$
f(x) = 
\begin{cases} 
\ln x & , x > 0 \\
a^x & , x \leq 0
\end{cases}
$$
where \( a > 0 \) and \( a \neq 1 \). If \( f(e^2) = f(-2) \), then the real number \( a = \) \_\_\_\_\_\_.

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