
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
Find the smallest  $\lambda \in \mathbb{R}$  such that for all  $n \in \mathbb{N}_+$ , there exists  $x_1, x_2, \ldots, x_n$  satisfying  $n = x_1 x_2 \ldots x_{2023}$ , where  $x_i$  is either a prime or a positive integer not exceeding  $n^\lambda$  for all  $i \in \left\{ 1,2, \ldots, 2023 \right\}$ .

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