
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
Reading comprehension question:  
Define: If the square of a number equals -1, denoted as $i^{2}=-1$, then this number $i$ is called the imaginary unit. We call numbers in the form of $a+bi$ (where $a$ and $b$ are real numbers) complex numbers. Here, $a$ is called the real part of the complex number, and $b$ is called the imaginary part. The addition, subtraction, and multiplication operations of complex numbers are similar to those of polynomials.  
For example, calculate:  
$(2-i) + (5+3i) = (2+5) + (-1+3)i = 7+2i$  
$(1+i) \times (2-i) = 1 \times 2 - 1 \times i + 2 \times i - i^{2} = 2 + (-1+2)i + 1 = 3+i$  
$i^{3} = i^{2} \times i = -1 \times i = -i$  
$i^{4} = i^{2} \times i^{2} = -1 \times (-1) = 1$  
Based on the above information, complete the following questions:  
$(1)$ Fill in the blank: $i^{6} =$ ______  
$(2)$ Calculate: $(1+i) \times (3-4i) + i^{5}$  
$(3)$ Calculate: $i+i^{2}+i^{3}+\cdots \cdots +i^{2020}+i^{2021}+i^{2022}+i^{2023}$.

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