
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 quadratic function $f(x)=mx^{2}-2x-3$, the solution set of the inequality $f(x)\leqslant 0$ with respect to the real number $x$ is $(-1,n)$  
$(1)$ When $a > 0$, solve the inequality with respect to $x$: $ax^{2}+n+1 > (m+1)x+2ax$;  
$(2)$ Does there exist a real number $a\in(0,1)$, such that the minimum value of the function $y=f(a^{x})-3a^{x+1}$ $(x\in[1,2])$ is $-5$? If it exists, find the value of the real number $a$; if not, explain why.

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