
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
Determine the number of ordered quintuples  $(a,b,c,d,e)$  of integers with  $0\leq a<$   $b<$   $c<$   $d<$   $e\leq 30$  for which there exist polynomials  $Q(x)$  and  $R(x)$  with integer coefficients such that \[x^a+x^b+x^c+x^d+x^e=Q(x)(x^5+x^4+x^2+x+1)+2R(x).\]


*Proposed by Michael Ren*

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