
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
In the ancient Chinese mathematical masterpiece "The Mathematical Classic of Sunzi" Volume $26$, the $26$th question is: "There is an unknown quantity, when divided by $3$, the remainder is $2$; when divided by $5$, the remainder is $3; when divided by $7$, the remainder is $2$. What is the quantity?" The mathematical meaning of this question is: find a positive integer that leaves a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $2$ when divided by $7$. This is the famous "Chinese Remainder Theorem". If we arrange all positive integers that leave a remainder of $2$ when divided by $3$ and all positive integers that leave a remainder of $2$ when divided by $7$ in ascending order to form sequences $\{a_{n}\}$ and $\{b_{n}\}$ respectively, and take out the common terms from sequences $\{a_{n}\}$ and $\{b_{n}\}$ to form sequence $\{c_{n}\}$, then $c_{n}=$____; if sequence $\{d_{n}\}$ satisfies $d_{n}=c_{n}-20n+20$, and the sum of the first $n$ terms of the sequence $\left\{\frac{1}{d_{n}d_{n+1}}\right\}$ is $S_{n}$, then $S_{2023}=$____.

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