
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 \(a + 2b + 3c + 4d + 5e = k\) and \(5a = 4b = 3c = 2d = e\), find the smallest positive integer value for \(k\) so that \(a, b, c, d\), and \(e\) are all positive integers.

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