
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 several numbers, one of them,  $a$ , is chosen and replaced by the three numbers  $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are  $1000$  ones, and we apply the process several times. A number  $m$  is called *good* if there are  $m$  or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.

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