
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  $V$  be a 10-dimensional real vector space and  $U_1,U_2$  two linear subspaces such that  $U_1 \subseteq U_2, \dim U_1 =3, \dim U_2=6$ . Let  $\varepsilon$  be the set of all linear maps  $T: V\rightarrow V$  which have  $T(U_1)\subseteq U_1, T(U_2)\subseteq U_2$ . Calculate the dimension of  $\varepsilon$ . (again, all as real vector spaces)

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