
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
Given that  $ 2^{2004}$  is a  $ 604$ -digit number whose first digit is  $ 1$ , how many elements of the set  $ S \equal{} \{2^0,2^1,2^2, \ldots,2^{2003}\}$  have a first digit of  $ 4$ ?

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