
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 $M$ is a non-empty subset of the set $\{1, 2, 3, \ldots, 2k-1\}$ (where $k \in \mathbb{N}^*$ and $k \geq 2$), and for any $x \in M$, it holds that $2k-x \in M$. Let the number of sets $M$ that satisfy this condition be denoted as $f(k)$. Then, find $f(2)$ and $f(k)$.

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