
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 \( x \) be a natural number. A sequence of natural numbers
\[ x_{0}=1, \quad x_{1}, \quad x_{2}, \cdots, \quad x_{l}=x \]
is called a factor chain of \( x \) if it satisfies the following conditions:
\[ x_{i-1}<x_{i}, \quad x_{i-1} \,|\, x_{i}, \quad i=1, 2, \cdots, l. \]

Here, \( l \) is the length of the factor chain. Let \( L(x) \) and \( R(x) \) represent the length of the longest factor chain of \( x \) and the number of such longest chains, respectively. For \( x = 5^k \cdot 31^m \cdot 1990^n \) (where \( k, m, n \) are natural numbers), find \( L(x) \) and \( R(x) \).

(The 5th Chinese National High School Mathematics Winter Camp, 1990)

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