
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
Square $ABCD$ is located on the plane coordinate system, where $A(1,1)$, $B(-1,1)$, $C(-1,-1)$, $D(1,-1)$. Consider three transformations on this square: $\left(1\right)L$: counterclockwise rotation by $90^{\circ}$. $\left(2\right)R$: clockwise rotation by $90^{\circ}$. $\left(3\right)S$: reflection about the origin. The above three operations can transform the square into itself, but the positions of points $A$, $B$, $C$, $D$ will change. For example, after applying transformation $R$ to the original square, vertex $A$ moves from $(1,1)$ to $(1,-1)$, and then after applying transformation $S$, $A$ moves to $(-1,1)$. Denote the sequence of $k$ transformations on the original square as $a_{1}a_{2}\cdots a_{k}$, where $a_{i}\in \{L,R,S\}$, $i=1,2,\cdots,k$. If after $k$ transformations, the positions of the vertices return to their original locations, we call such a transformation a $k-$identity transformation. For example, $RRS$ is a $3-$identity transformation. There are _______ $3-$identity transformations in total; for a positive integer $n$, there are _______ $n-$identity transformations in total.

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