
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 every positive integer  $M \geq  2$ , find the smallest real number  $C_M$  such that for any integers  $a_1, a_2,\ldots , a_{2023}$ , there always exist some integer  $1 \leq k < M$  such that

\[\left\{\frac{ka_1}{M}\right\}+\left\{\frac{ka_2}{M}\right\}+\cdots+\left\{\frac{ka_{2023}}{M}\right\}\leq C_M.\]
Here,  $\{x\}$  is the unique number in the interval  $[0, 1)$  such that  $x - \{x\}$  is an integer.
*Proposed by usjl*

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