
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
Find the positive integer  $N$  for which there exist reals  $\alpha, \beta, \gamma, \theta$  which obey
\begin{align*}
0.1 &= \sin \gamma \cos \theta \sin \alpha, 
0.2 &= \sin \gamma \sin \theta \cos \alpha, 
0.3 &= \cos \gamma \cos \theta \sin \beta, 
0.4 &= \cos \gamma \sin \theta \cos \beta, 
0.5 &\ge \left\lvert N-100 \cos2\theta \right\rvert.
\end{align*}*Proposed by Evan Chen*

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