
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 interval $[1, +\infty)$ that satisfies: (1) $f(2x) = cf(x)$ ($c$ is a positive constant); (2) when $2 \leq x \leq 4$, $f(x) = 1 - |x - 3|$. If all the local maximum points of the function fall on the same line, then $c = \boxed{1 \text{ or } 2}$.

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