
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
Let  $ f$  be a function defined on the set of non-negative integers and taking values in the same
set. Given that

(a)  $ \displaystyle x \minus{} f(x) \equal{} 19\left[\frac{x}{19}\right] \minus{} 90\left[\frac{f(x)}{90}\right]$  for all non-negative integers  $ x$ ;

(b)  $ 1900 < f(1990) < 2000$ ,

find the possible values that  $ f(1990)$  can take.
(Notation : here  $ [z]$  refers to largest integer that is  $ \leq z$ , e.g.  $ [3.1415] \equal{} 3$ ).

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