
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 PUMaC grader is grading the submissions of forty students  $s_1, s_2, ..., s_{40}$  for the individual finals round, which has three problems. After grading a problem of student  $s_i$ , the grader either: $\bullet$   grades another problem of the same student, or $\bullet$  grades the same problem of the student  $s_{i-1}$  or  $s_{i+1}$  (if  $i > 1$  and  $i < 40$ , respectively).
He grades each problem exactly once, starting with the first problem of  $s_1$  and ending with the third problem of  $s_{40}$ . Let  $N$  be the number of different orders the grader may grade the students’ problems in this way. Find the remainder when  $N$  is divided by  $100$ .

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