
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 the function \( f(x)=\sin x+\sqrt{3} \cos x+1 \):

(1) Find the maximum and minimum values of the function \( f(x) \) on the interval \(\left[0, \frac{\pi}{2}\right]\).

(2) Given that there exist real numbers \( a \), \( b \), and \( c \) such that \( a f(x)+b f(x-c)=1 \) holds for any \( x \in \mathbf{R} \), find the value of \( \frac{b \cos c}{a} \).

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