
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
Denote by \(\langle x\rangle\) the fractional part of the real number \(x\) (for instance, \(\langle 3.2\rangle = 0.2\)). A positive integer \(N\) is selected randomly from the set \(\{1, 2, 3, \ldots, M\}\), with each integer having the same probability of being picked, and \(\left\langle\frac{87}{303} N\right\rangle\) is calculated. This procedure is repeated \(M\) times and the average value \(A(M)\) is obtained. What is \(\lim_{M \rightarrow \infty} A(M)\)?

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