
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 that the function $f(x)$ is an odd function defined on $[-3,3]$, and when $0 \lt x\leqslant 3$, $f(x)=\frac{1}{2}x^2+x$. Find:<br/>
$(1)$ The analytical expression of the function $f(x)$ when $-3\leqslant x \lt 0$;<br/>
$(2)$ If $f(a+1)+f(2a-1) \gt 0$, find the range of real number $a$.

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