
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  $m$  be a positive integer and  $a$  a positive real number. Find the greatest value of  $a^{m_1}+a^{m_2}+\ldots+a^{m_p}$  where  $m_1+m_2+\ldots+m_p=m, m_i\in\mathbb{N},i=1,2,\ldots,p;$   $1\leq p\leq m, p\in\mathbb{N}$ . 

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