
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 $a\in \mathbb{R}$, $f\left(x\right)=ax^{2}+2x-3$.  
$(1)$ For the equation $f\left(x\right)=0$ to have two positive roots, find the range of real number $a$;  
$(2)$ Solve the inequality $f\left(x\right) \gt 0$.

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