
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
Let the function be defined as
$$
f(x)=|\cos x+\alpha \cos 2 x+\beta \cos 3 x|,
$$
where $\alpha$ and $\beta$ are real numbers. Find 
$$
M=\min _{\alpha, \beta} \max _{x \in \mathbf{R}}\{f(x)\}.

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