
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 $f(x)=\ln(ax)-\frac{1}{3}x^3(a≠0)$.<br/>$(Ⅰ)$ When $a=2$, find the equation of the tangent line to the curve $y=f\left(x\right)$ at the point $(\frac{1}{2},f(\frac{1}{2}))$;<br/>$(Ⅱ)$ Discuss the monotonicity of the function $f\left(x\right)$;<br/>$(Ⅲ)$ When $a=1$, let $g\left(x\right)=f\left(x\right)+t$. If $g\left(x\right)$ has two distinct zeros, find the range of the parameter $t$.

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