
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
Let $A_n = \{x \,|\, 2^n < x < 2^{n+1}, x = 3m, m \in \mathbb{N}\}$. If $|A_n|$ denotes the number of elements in the set $A_n$, then find $|A_5|$ and $\sum_{k=1}^{10}|A_k|$.

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