
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 a sequence $\{a_n\}$ satisfying $a_1 = 1$, $\left| a_n - a_{n-3} \right| = \frac{1}{2^n} \quad (n \geq 2, \ n \in \mathbb{N})$, and the subsequence $\{a_{2n-1}\}$ is decreasing, while the subsequence $\{a_{2n}\}$ is increasing, find the value of $5 - 6a_{10}$.

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