
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 $f(x)= \begin{cases}
x^{2}+3ax+a^{2}-3 & (x < 0) \\
2e^{x}-(x-a)^{2}+3 & (x > 0) \end{cases}$, with $a\in\mathbb{R}$.  
(1) If the function $y=f(x)$ attains an extreme value at $x=1$, find the value of $a$.  
(2) If there exist two points on the graph of the function $y=f(x)$ that are symmetric with respect to the origin, determine the range of $a$.

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