
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
Numbers  $x,y,z$  are positive integers and satisfy the equation  $x+y+z=2013$ . (E)
a) Find the number of the triplets  $(x,y,z)$  that are solutions of the equation (E).
b) Find the number of the solutions of the equation (E) for which  $x=y$ .
c) Find the solution  $(x,y,z)$   of the equation (E) for which the product  $xyz$  becomes maximum.

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