
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
Do there exist two sequences of real numbers  $ \{a_i\}, \{b_i\},$   $ i \in \mathbb{N},$  satisfying the following conditions:

\[ \frac{3 \cdot \pi}{2} \leq a_i \leq b_i\]

and

\[ \cos(a_i x) \minus{} \cos(b_i x) \geq \minus{} \frac{1}{i}\]

 $ \forall i \in \mathbb{N}$  and all  $ x,$  with  $ 0 < x < 1?$ 

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