
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  $T=\frac{1}{4}x^{2}-\frac{1}{5}y^{2}+\frac{1}{6}z^{2}$  where  $x,y,z$  are real numbers such that  $1 \leq x,y,z \leq 4$  and  $x-y+z=4$ .
Find the smallest value of  $10 \times T$ .

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