
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
A four-digit number  $n$  is said to be *literally 1434* if, when every digit is replaced by its remainder when divided by  $5$ , the result is  $1434$ . For example,  $1984$  is *literally 1434* because  $1$  mod  $5$  is  $1$ ,  $9$  mod  $5$  is  $4$ ,  $8$  mod  $5$  is  $3$ , and  $4$  mod  $5$  is  $4$ . Find the sum of all four-digit positive integers that are *literally 1434*.

*Proposed by Evin Liang*

<details><summary>Solution</summary>*Solution.*  $\boxed{67384}$ The possible numbers are  $\overline{abcd}$  where  $a$  is  $1$  or  $6$ ,  $b$  is  $4$  or  $9$ ,  $c$  is  $3$  or  $8$ , and  $d$  is  $4$  or  $9$ . There are  $16$  such numbers and the average is  $\dfrac{8423}{2}$ , so the total in this case is  $\boxed{67384}$ .</details>

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