
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 domain of the function $f(x)$ is $D$, and the set $M \subseteq D$, if there exists a nonzero real number $t$ such that for all $x \in M$, $x-t \in D$, and $f(x-t) < f(x)$, then $f(x)$ is called a $t\oplus$ function on $M$.  
$(1)$ Given functions $g(x)=x$ and $h(x)=\frac{1}{x}$, determine whether $g(x)$ and $h(x)$ are $3\oplus$ functions on the interval $[4,6]$, and explain the reasons.  
$(2)$ Given the function $f(x)=x+\frac{1}{x}$, and $f(x)$ is an $n\oplus$ function on the interval $[-2,-1]$, find the smallest positive integer value of $n$.

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