
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
A finite sequence of integers  $a_1, a_2, \dots, a_n$  is called *regular* if there exists a real number  $x$  satisfying \[ \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. \] Given a regular sequence  $a_1, a_2, \dots, a_n$ , for  $1 \le k \le n$  we say that the term  $a_k$  is *forced* if the following condition is satisfied: the sequence \[ a_1, a_2, \dots, a_{k-1}, b \] is regular if and only if  $b = a_k$ .  Find the maximum possible number of forced terms in a regular sequence with  $1000$  terms.

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