
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 positive integer \( n \) and a real number \( x \) such that \( 0 \leqslant x < n \), define

\[ f(n, x) = (1 - \{x\}) \cdot \binom{n}{\lfloor x \rfloor} + \{x\} \cdot \binom{n}{\lfloor x \rfloor + 1} \]

where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to the real number \( x \), and \( \{x\} = x - \lfloor x \rfloor \). If integers \( m, n \geq 2 \) satisfy

\[ f\left(m, \frac{1}{n}\right) + f\left(m, \frac{2}{n}\right) + \cdots + f\left(m, \frac{mn - 1}{n}\right) = 123, \]

find the value of

\[ f\left(n, \frac{1}{m}\right) + f\left(n, \frac{2}{m}\right) + \cdots + f\left(n, \frac{mn - 1}{m}\right). \]

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