
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
For a real number  $ x$  in the interval  $ (0,1)$  with decimal representation 

\[ 0.a_1(x)a_2(x)...a_n(x)...,\] 

denote by  $ n(x)$  the smallest nonnegative integer such that 

\[ \overline{a_{n(x)\plus{}1}a_{n(x)\plus{}2}a_{n(x)\plus{}3}a_{n(x)\plus{}4}}\equal{}1966 .\] 

Determine  $ \int_0^1n(x)dx$ . ( $ \overline{abcd}$  denotes the decimal number with digits  $ a,b,c,d .$ ) 

*A. Renyi*

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