
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
There are 1000 doors  $D_1, D_2, . . . , D_{1000}$  and 1000 persons  $P_1, P_2, . . . , P_{1000}$ .
Initially all the doors were closed. Person  $P_1$  goes and opens all the doors.
Then person  $P_2$  closes door  $D_2, D_4, . . . , D_{1000}$   and leaves the odd numbered doors open. Next  $P_3$  changes the state of every third door, that
is,  $D_3, D_6, . . . , D_{999}$  . (For instance,  $P_3$  closes the open door  $D_3$  and opens
the closed door D6, and so on). Similarly,  $P_m$  changes the state of the
the doors  $D_m, D_{2m}, D_{3m}, . . . , D_{nm}, . . .$  while leaving the other doors untouched. Finally,  $P_{1000}$  opens  $D_{1000}$  if it was closed or closes it if it were
open. At the end, how many doors will remain open?

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