
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 positive integer \( n \) is magical if
$$
\lfloor\sqrt{\lceil\sqrt{n}\rceil}\rfloor = \lceil\sqrt{\lfloor\sqrt{n}\rfloor}\rceil
$$
where \( \lfloor\cdot\rfloor \) and \( \lceil\cdot\rceil \) represent the floor and ceiling function respectively. Find the number of magical integers between 1 and 10,000, inclusive.

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