
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 \( N \) be the set of non-negative integers. The function \( f: N \to N \) satisfies \( f(a+b) = f(f(a)+b) \) for all \( a \), \( b \) and \( f(a+b) = f(a) + f(b) \) for \( a+b < 10 \). Also \( f(10) = 1 \). How many three-digit numbers \( n \) satisfy \( f(n) = f(N) \), where \( N \) is the "tower" 2, 3, 4, 5, in other words, it is \( 2^{a} \), where \( a = 3^{b} \), where \( b = 4^{5} \)?

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