
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  $ c$  be a positive integer. The sequence  $ a_1,a_2,\ldots$  is defined as follows  $ a_1\equal{}c$ ,  $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$  for all positive integers  $ n$ . Find all  $ c$  so that there are integers  $ k\ge1$  and  $ m\ge2$  so that  $ a_k^2\plus{}c^3$  is the  $ m$ th power of some integer.

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