
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 function $f(x)=2x+\ln x-a(x^{2}+x)$.  
$(1)$ If the tangent line of the function $f(x)$ at $x=1$ is parallel to the line $y=-3x$, find the value of the real number $a$;  
$(2)$ If there exists $x\in(0,+\infty)$ such that the inequality $f(x)\geqslant 0$ holds, find the range of values for the real number $a$;  
$(3)$ When $a=0$, let the function $p(x)=2x+1-f(x)$, $q(x)=x^{3}-mx+e$ (where $e$ is the base of the natural logarithm, and $m$ is a parameter). Denote the function $h(x)= \dfrac {p(x)+q(x)+|p(x)-q(x)|}{2}$. Determine the number of zeros of the function $h(x)$.

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