
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 quadratic function $f(x)=x^{2}+2ax+2$.  
$(1)$ If the inequality $f(x) > 3ax$ holds for $1\leqslant x\leqslant 5$, find the range of real numbers for $a$.  
$(2)$ Solve the inequality in terms of $x$: $\left(a+1\right)x^{2}+x > f(x)$ (where $a\in \mathbb{R}$).

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