
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 functions $f(x)=alnx+\frac{1}{2}x^2$ and $g(x)=(a+1)x$ ($a$ is a constant).  
$(1)$ Discuss the monotonicity of $f(x)$.  
$(2)$ If the inequality $f(x) \leq g(x)$ holds for all $x \in [1,4]$, find the range of real number $a$.

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