
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 two complex numbers $z_{1} = \sin^{2}x + i\cdot\cos2x$ and $z_{2} = \sin^{2}x + i\cdot\cos x$ (where $x\in\mathbb{R}$ and $i$ is the imaginary unit), determine whether $z_1$ and $z_2$ can represent the same point on the complex plane. If yes, specify the complex number corresponding to that point; if no, provide a reason.

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