
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
Find the number of numbers \( N \) from the set \(\{1, 2, \ldots, 2018\}\) for which positive solutions \( x \) exist for the equation

$$
x^{[x]} = N
$$

(where \([x]\) is the integer part of the real number \( x \), i.e., the largest integer not exceeding \( x \)).

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