
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
How many  $5$ -digit numbers  $N$  (in base  $10$ ) contain no digits greater than  $3$  and satisfy the equality  $\gcd(N,15)=\gcd(N,20)=1$ ? (The leading digit of  $N$  cannot be zero.)

*Based on a proposal by Yannick Yao*

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