
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
Find the number of 5-digit numbers that include at least one '6' and are divisible by 3. For an \( n \)-digit number \( a_{n-1}a_{n-2} \cdots a_1a_0 \):

\[ a_{n-1}a_{n-2} \cdots a_1a_0 \equiv a_k \times 10^{k-1} + a_{k-2} \times 10^{k-2} + \cdots + a_1 \times 10 + a_0 \]

Since \( 10 \equiv 1 \pmod{3}, 10^2 \equiv 1 \pmod{3}, \cdots \), \( 10^k \equiv 1 \pmod{3} \), a necessary condition for \( a_{n-1}a_{n-2} \cdots a_1a_0 \) to be divisible by 3 is:

\[ a_{n-1} + a_{n-2} + \cdots + a_1 + a_0 \equiv 0 \pmod{3} \]

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