
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 \( f(x) \) be a given convex function defined on the interval \( 0 \leq x \leq 1 \) (i.e., for any \( 0 \leq x_{1} < x_{2} \leq 1 \) and \( 0 \leq \lambda \leq 1 \), the inequality

\[ f(\lambda x_{1} + (1 - \lambda) x_{2}) \leq \lambda f(x_{1}) + (1 - \lambda) f(x_{2}) \]

holds true). Determine the locus of points \( P(u, v) \) in the plane for which the system of equations

\[ f(\sin^{2} x) + f(\cos^{2} y) = u \]

\[ \sin^{2} x + \cos^{2} y = v \]

has a solution.

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