
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 \( f(x) \) is defined on \(\mathbf{R}^{+}\) and satisfies the following two conditions:
(1) \( f(x) \) is monotonically decreasing on \(\mathbf{R}^{+}\) and \( f(x) > \frac{1}{x^{2}} \);
(2) It always holds that \( f^{2}(x) \cdot f\left[f(x) - \frac{1}{x^{2}}\right] = f^{3}(1) \) on \(\mathbf{R}^{+}\),

(1) Find the value of \( f(1) \);
(2) Provide a function \( f(x) \) that satisfies the conditions and verify it.

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