
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
Observe the following equations:<br/>The $1$st equation: $a_1=\frac{1}{1×3}=\frac{1}{2}×(1-\frac{1}{3})$,<br/>The $2$nd equation: $a_2=\frac{1}{3×5}=\frac{1}{2}×(\frac{1}{3}-\frac{1}{5})$,<br/>The $3$rd equation: $a_3=\frac{1}{5×7}=\frac{1}{2}×(\frac{1}{5}-\frac{1}{7})$,<br/>The $4$th equation: $a_4=\frac{1}{7×9}=\frac{1}{2}×(\frac{1}{7}-\frac{1}{9})$,<br/>$\ldots $<br/>Please answer the following questions:<br/>$(1)$ According to the above pattern, write out the $5$th equation: $a_{5}=\_\_\_\_\_\_=\_\_\_\_\_\_$;<br/>$(2)$ Find the value of $a_{1}+a_{2}+a_{3}+\ldots +a_{2023}$.

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