
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
Given \( a, b, c \in \mathbf{R}_{+} \) and \( a^{2} + b^{2} + c^{2} = 1 \), let
\[ M = \max \left\{ a + \frac{1}{b}, b + \frac{1}{c}, c + \frac{1}{a} \right\}, \]
find the minimum value of \( M \).

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