
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
The equation of line  $\ell_1$  is  $24x-7y = 319$  and the equation of line  $\ell_2$  is  $12x-5y = 125$ . Let  $a$  be the number of positive integer values  $n$  less than  $2023$  such that for both  $\ell_1$  and  $\ell_2$  there exists a lattice point on that line that is a distance of  $n$  from the point  $(20,23)$ . Determine  $a$ .

*Proposed by Christopher Cheng*

<details><summary>Solution</summary>*Solution.*  $\boxed{6}$ Note that  $(20,23)$  is the intersection of the lines  $\ell_1$  and  $\ell_2$ . Thus, we only care about lattice points on the the two lines that are an integer distance away from  $(20,23)$ . Notice that  $7$  and  $24$  are part of the Pythagorean triple  $(7,24,25)$  and  $5$  and  $12$  are part of the Pythagorean triple  $(5,12,13)$ . Thus, points on  $\ell_1$  only satisfy the conditions when  $n$  is divisible by  $25$  and points on  $\ell_2$  only satisfy the conditions when  $n$  is divisible by  $13$ . Therefore,  $a$  is just the number of positive integers less than  $2023$  that are divisible by both  $25$  and  $13$ . The LCM of  $25$  and  $13$  is  $325$ , so the answer is  $\boxed{6}$ .</details>

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