
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
If the quadratic function $f(x)$ satisfies $f(x+1)+f(x)=-x^{2}-5x-\frac{5}{2}$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ If the function $g(x)=x\ln x+f(x)$, solve the inequality with respect to $x$: $g(x^{2}+x)\geq g(2)$.

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