
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
Define set $T = \{3^k | k \in \mathbb{Z}, 0 \le k \le 1000 \}$. Given that $3^{1000} = 5050 \cdots 0497$ contains $477$ digits, determine how many elements of $T$ begin with the digit $1$.

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