
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
How many ordered triples  $(a,b,c)$  of integers satisfy the inequality \[a^2+b^2+c^2 \leq a+b+c+2?\]

Let  $T = TNYWR$ . David rolls a standard  $T$ -sided die repeatedly until he first rolls  $T$ , writing his rolls in order on a chalkboard. What is the probability that he is able to erase some of the numbers he's written such that all that's left on the board are the numbers  $1, 2, \dots, T$  in order?


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