
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
Determine the least constant \( M \) such that
\[ 
\frac{x_{1}}{x_{1}+x_{2}}+\frac{x_{2}}{x_{2}+x_{1}}+\frac{x_{3}}{x_{3}+x_{4}}+\cdots+\frac{x_{2009}}{x_{2009}+x_{1}} < M 
\]
for any positive real numbers \( x_{1}, x_{2}, x_{3}, \ldots, x_{2009} \).

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