
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 $a, c \in \mathbb{R}$, the function $f(x) = \frac{x^2 + (3a+1)x + c}{x+a}$.  
$(1)$ If $a=0$, find the domain of the function and determine if there exists a value of $c$ such that $f(x)$ is an odd function, and explain the reason.  
$(2)$ If the function passes through the point $(1,3)$ and intersects the negative half-axis of the $x$-axis at two distinct points, find the values of $c$ and the range of $a$ in this case.

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