
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
There is a grid of height 2 stretching infinitely in one direction. Between any two edge-adjacent cells of the grid, there is a door that is locked with probability \(\frac{1}{2}\), independent of all other doors. Philip starts in a corner of the grid (in the starred cell). Compute the expected number of cells that Philip can reach, assuming he can only travel between cells if the door between them is unlocked.

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