
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
Define a function \( f(x) \) on the set of positive integers such that \( f(1) = \frac{4}{3} \) and \( 2 f(k) = f(k-1) - k - 5^k + 2^k \) for \( k \geq 2, k \in \mathbb{N} \). Find the explicit formula for the function \( f(x) \).

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