
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 that $\theta$ is a real number, if the complex number $z=\sin 2\theta-1+i(\sqrt{2}\cos\theta-1)$ is a purely imaginary number, then the imaginary part of $z$ is \_\_\_\_\_.

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